aa r X i v : . [ m a t h . N T ] F e b CLASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS
DANIEL C. MAYER
Abstract.
The number of non-isomorphic cubic fields L sharing a common discriminant d L = d is called the multiplicity m = m ( d ) of d . For an assigned value of d , these fields are collected ina multiplet M d = ( L , . . . , L m ). In this paper, the information in all existing tables of totallyreal cubic number fields L with positive discriminants d L < is extended by computing thedifferential principal factorization types τ ( L ) ∈ { α , α , α , β , β , γ, δ , δ , ε } of the members L of each multiplet M d of non-cyclic fields, a new kind of arithmetical invariants which providesuccinct information about ambiguous principal ideals and capitulation in the normal closures N of non-Galois cubic fields L . The classification is arranged with respect to increasing 3-class rank ̺ of the quadratic subfields K of the S -fields N , and to ascending number of prime divisors ofthe conductor f of N/K . The Scholz conjecture concerning the distinguished index of subfieldunits ( U N : U ) = 1 for ramified extensions N/K with conductor f > Introduction
Recently, we have classified all complex and totally real cubic number fields L with discriminantsin the range −
20 000 < d L <
100 000, covered in the years between 1972 and 1976 by Angell [1, 2].The reconstruction [35] was carried out from the viewpoint of 3-ring class fields with Magma [25].In this paper, we omit simply real cubic fields with a few types only, and we rather put ourfocus on triply real cubic fields L with nine possible types τ ( L ) ∈ { α , α , α , β , β , γ, δ , δ , ε } ,which refine the coarse classification with five types α, β, γ, δ, ε by Moser [36]. According to thehistorical development of systematic investigations of triply real cubic fields, we present our refinedclassification of multiplets of totally real cubic fields L in four steps with increasing upper bounds100 000 ,
200 000 ,
500 000, and 10 000 000 for the discriminants d L .In §
4, we start by recalling our results in [35] concerning the range d L <
100 000 of Angell[1, 2]. In §
5, we continue with an update of our extension to d L <
200 000 in [26, 27], which wascomputed in 1991 by means of the Voronoi algorithm [40]. Whereas the count of discriminantsand fields and the collection of fields into multiplets were correct, the classification into type δ instead of β was partially erroneous, because absolute principal factors do not necessarily showup among the lattice minima in the chains of Voronoi’s algorithm. The coronation of this paperwill be established in §§ d L <
500 000 of Ennola and Turunen [9, 10] and the range d L <
10 000 000 ofLlorente and Quer [24], the most extensive ranges deposited in files of unpublished mathematicaltables (UMT). Statistical evaluations and theoretical conclusions are given in § counting cubic fields even inbigger ranges, without computation of arithmetical invariants , like fundamental systems of unitsand class group structures, and without classification into differential principal factorization types .The latter are introduced in §
3, where we prove that the F p -vector space P N/K / P K of primitiveambiguous principal ideals of a number field extension N/K with odd prime degree p can beendowed with a natural trichotomic direct product structure.Finally, as an application of the notions of multiplets and DPF types, the Scholz conjecture [39]concerning the normal closure N of L is stated, refined, and proved completely in § Mathematics Subject Classification. Primary 11R37, 11R11, 11R16, 11R20, 11R27, 11R29, 11Y40. Key words and phrases. S -fields, dihedral fields, multiplicity of discriminants, 3-Selmer space, 3-ring spaces, multiplets,Galois cohomology, differential principal factorizations, capitulation of 3-class groups, statistics, Scholz conjecture.Research supported by the Austrian Science Fund (FWF): projects J0497-PHY and P26008-N25. Quer, which did not illuminate the constitution of multiplets ( L , . . . , L m ) of totally real cubicfields as subfields of a common 3-ring class field K f modulo a 3-admissible conductor f over aquadratic base field K , let alone the differential principal factorization types ( τ ( L ) , . . . , τ ( L m )).Secondly, it is our intention to show the increasing wealth of arithmetical structure in three successiveextensions of the upper bound B from 100 000 to 200 000, 500 000, and finally 10 000 000. Therearise conductors f divisible by an ascending number of primes, new types τ ( L ), and heterogeneous multiplets M ( K f ) = [ M c d ] c | f = [( L c, , . . . , L c,m ( c ) )] c | f with increasing complexity.2. Construction as subfields of a ring class field Structure and multiplicity of cubic discriminants. Let K be a quadratic number fieldwith fundamental discriminant d = d K (square free, except possibly for the 2-contribution). Definition 1. A positive integer f is called a 3- admissible conductor for K , if it has the shape f = 3 e · q · · · q t with an integer exponent e ∈ { , , } , t ≥ 0, and pairwise distinct prime numbers q , . . . , q t ∈ P \ { } , such that the following conditions are satisfied:Kronecker symbol (cid:18) dq i (cid:19) ≡ q i (mod 3) , for all 1 ≤ i ≤ t, and e ∈ { , } if d ≡ ± , { , } if d ≡ +3 (mod 9) , { , , } if d ≡ − . So, a 3-admissible conductor f for K is essentially square free, except possibly for the critical contribution by the prime 3. The condition involving the Kronecker symbol means that a non-critical prime divisor q = 3 of f must remain inert in K , if q ≡ − K ,if q ≡ +1 (mod 3). The critical prime divisor 3 of f is 3-admissible if and only if it ramifies in K ,that is, if d ≡ ± f is 3-admissible.(Recall that 3 remains inert in K if d ≡ − K if d ≡ +1 (mod 3).) So far,all contributions to f are regular. There is, however, the possibility of an irregular f , when d ≡ − Definition 2. An integer D = f · d is called a formal cubic discriminant if f is a 3-admissibleconductor for the quadratic field K with fundamental discriminant d . (Since the square f andthe fundamental discriminant d are congruent to 0 or 1 modulo 4, this is also the case for a formalcubic discriminant D . We shall see that D is not necessarily discriminant d L of a cubic field L .)Note that this definition does not include discriminants d L of cyclic cubic fields L which are perfectsquares f of conductors f exactly divisible by primes congruent to 1 modulo 3, and possibly alsoby the prime power 3 . In order to determine the multiplicity of d L , we need further definitions. Definition 3. An algebraic number α = 0 in the quadratic field K is called a 3- virtual unit , if itsprincipal ideal α O K is the cube j of an ideal j of K . Obviously all units η in U K and all thirdpowers α = 0 in ( K × ) are 3-virtual units of K . Let I denote the group of all 3-virtual units of K ,and let K × = K \ { } denote the multiplication group of K . The F -vector space V := I/ ( K × ) is called the 3- Selmer space of K .For any positive integer n and a set X of algebraic numbers, let X ( n ) be the subset of X consistingof elements coprime to n . The 3-Selmer space V of K is isomorphic to the direct product of the3-elementary class group Cl K / Cl K and the 3-elementary unit group U K /U K of K [31, p. 2212].Since any ideal class contains an ideal coprime to an assigned positive integer n , it follows thatCl K = I K / P K ≃ I K ( n ) / P K ( n ), and since trivially U K = U K ( n ), we have V ≃ I ( n ) /K ( n ) [31,Dfn. 2.2, p. 2211]. Definition 4. Let f be a positive integer. Denote by S f := { α ∈ K | α ≡ f ) } the ray modulo f of K , and by R f := Q ( f ) · S f the ring modulo f of K . The subspace V ( f ) :=( I ( f ) ∩ R f · K ( f ) ) /K ( f ) of the 3-Selmer space V is called the 3- ring space modulo f of K . Itscodimension δ ( f ) := codim( V ( f )) = dim F ( V /V ( f )) is called the 3- defect of f with respect to K . LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 3 In order to enable comparison and binary operations (in particular, intersection) of two differentring spaces V ( f ) and V ( f ′ ), we need the concept of a modulus of declaration , that is a positiveinteger n which is a common multiple of f and f ′ . Then V ( f ) ≃ ( I ( n ) ∩ R f · K ( n ) ) /K ( n ) and V ( f ′ ) ≃ ( I ( n ) ∩ R f ′ · K ( n ) ) /K ( n ) , whence it makes sense to speak about inclusion and meet. Remark 1. It is possible to avoid the requirement of a modulus of declaration, if the theory ofring spaces is based on the approach via id`ele groups . This has been done by Satg´e [37] for primeconductors f = q and will be expanded further by ourselves for any f in a future paper.2.2. Homogeneous and heterogeneous multiplets. If f is a 3-admissible conductor with 3-defect δ ( f ) for a quadratic field K with fundamental discriminant d and 3-class rank ̺ , then thesum of all multiplicities m ( D ) of formal cubic discriminants D = c d with c running over all divisorsof f is given by X c | f m ( c d ) = 12 (3 ̺ f − ring class rank modulo f of K [31, Thm. 2.1, p. 2213], ̺ f = ̺ + t + w − δ ( f ) , where t := { q ∈ P \ { } | v q ( f ) = 1 } , and w is defined in terms of the 3-valuation v ( f ) of f , w := v ( f ) = 0 , v ( f ) = 1 or [ v ( f ) = 2 and d ≡ ± , v ( f ) = 2 and d ≡ . Definition 5. Let f be a 3-admissible conductor for a quadratic field K .(1) For each divisor c of f which is also a 3-admissible conductor for K , the multiplet M c d := ( L c, , . . . , L c,m ) with m = m ( c d )is called the homogeneous multiplet of cubic fields L c,i with discriminant c d .(2) The multiplet M ( K f ) := [ M c d ] c | f is called the heterogeneous multiplet of the 3-ring classfield K f modulo f of K . (The normal closures of all cubic fields L c,i with c | f and1 ≤ i ≤ m ( c d ) are subfields of the ring class field K f .)(3) The family sgn( M ( K f )) := [ m ( c d )] c | f of all partial multiplicities associated with f is calledthe signature of the heterogeneous multiplet M ( K f ). D = c d is only a formal but not an actual cubic discriminant if and only if the multiplicity m ( c d ) = 0 vanishes, that is, if M c d = ∅ is a nilet (denoted by the empty set symbol). Definition 6. By the type of the -ring class field K f modulo f of K we understand the followingpair (Obj( K f ) , Inv( K f )) of heterogeneous multiplets(2.1) Obj( K f ) := M ( K f ) = [( L c,i ) ≤ i ≤ m ( c d ) ] c | f Inv( K f ) := τ ( M ( K f )) = [( τ ( L c,i )) ≤ i ≤ m ( c d ) ] c | f consisting of all non-cyclic cubic fields L c,i with discriminants c d dividing f d as objects and theirdifferential principal factorization types τ ( L c,i ) as invariants .(See [33, 34] and the next section § Algorithmic process of construction. The computational technique which will be em-ployed for the construction of totally real cubic fields in the sections §§ 4, 5, 6, and 7 consists of twosteps. For an assigned real quadratic field K with fundamental discriminant d and a 3-admissibleconductor f for K , initially all cyclic cubic extensions N/K with conductor f are constructed assubfields of the ray class field modulo f of K . Then the members N of this family are tested fortheir absolute automorphism group G = Gal( N/ Q ), and only those with G ≃ S are permitted topass the filter. As a double check, we additionally make sure that the non-Galois subfields L < N have the required discriminant d L = f · d , and thus N is subfield of K f , the ring class field modulo f of K , which is contained in the ray class field modulo f of K . The result is the multiplet M f d ,because the fields N are certainly not subfields of K c for proper divisors c of f .Before we apply this algorithm, however, we have to introduce the concept of differential principalfactorizations (DPF) in section § DANIEL C. MAYER Differential principal factorization types Our intention in this section is to prepare sound foundations for the concept of differential principalfactorization (DPF) types and to establish a common theoretical framework for the classification • of dihedral fields N/ Q of degree 2 p with an odd prime p , viewed as subfields of suitable p -ring class fields over a quadratic field K (see the left part of Figure 1), and • of pure metacyclic fields N = K ( p √ D ) of degree ( p − · p with an odd prime p , viewed asKummer extensions of a cyclotomic field K = Q ( ζ p ) (see the right part of Figure 1),by the following arithmetical invariants:(1) the F p -dimensions of subspaces of the space P N/K / P K of primitive ambiguous principalideals, which are also called differential principal factors , of N/K ,(2) the capitulation kernel ker( T N/K ) of the transfer homomorphism T N/K : Cl p ( K ) → Cl p ( N ), a P K ( a O N ) P N , of p -classes from K to N , and(3) the Galois cohomology ˆH ( G, U N ), H ( G, U N ) of the unit group U N as a module over thecyclic automorphism group G = Gal( N/K ) ≃ C p . Figure 1. Dihedral and metacyclic situation ✉ Q rational field ✟✟✟✟✟✟ [ K : Q ] = 2 ✉ K = Q ( √ d )quadratic field[ L : Q ] = p cyclic extension ❡ LL , . . . , L p − p conjugates ✟✟✟✟✟✟ ✉ N = L · K dihedral fieldof degree 2 p ✉ Q rational field ✟✟✟✟✟✟ [ K : Q ] = p − ✉ ✉ K = Q ( ζ p )cyclotomic field[ L : Q ] = p ✻ intermediatefields ❄ Kummer extension ❡ L = Q ( p √ D ) L , . . . , L p − p conjugates ✟✟✟✟✟✟ ❡ ✉ N = L · K metacyclic fieldof degree ( p − p Primitive ambiguous ideals. Let p ≥ N/K be a relative extensionof number fields with degree p ( not necessarily Galois). Definition 7. The group I N of fractional ideals of N contains the subgroup of ambiguous ideals of N/K , denoted by the symbol I N/K := { A ∈ I N | A p ∈ I K } . The quotient I N/K / I K is calledthe F p - vector space of primitive ambiguous ideals of N/K . (Cfr. [33, Dfn. 3.1, p. 1991].) Proposition 1. Let L , . . . , L t be the totally ramified prime ideals of N/K , then a basis and thedimension of the space I N/K / I K over F p are finite and given by (3.1) I N/K / I K ≃ t Y i =1 ( h L i i / h L pi i ) ≃ F tp , dim F p ( I N/K / I K ) = t, whereas I N/K is an infinite abelian group containing I K .Proof. According to the definition of I N/K , the quotient I N/K / I K is an elementary abelian p -group. By the decomposition law for prime ideals of K in N , the space I N/K / I K is generated bythe totally ramified prime ideals (with ramification index e = p ) of N/K , that is to say I N/K = h L ∈ P N | L p ∈ P K iI K . According to the theorem on prime ideals dividing the discriminant, thenumber t of totally ramified prime ideals L , . . . , L t of N/K is finite . (cid:3) If L is another subfield of N such that N = L · K is the compositum of L and K , and N/L is ofdegree q coprime to p , then the relative norm homomorphism N N/L induces an epimorphism (3.2) N N/L : I N/K / I K → I L/F / I F , where F := L ∩ K denotes the intersection of L and K in Figure 2. Thus, by the isomorphismtheorem (see also [33, Thm. 4.2, pp. 1995–1996]), we have proved: LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 5 Theorem 1. There are the following two isomorphisms between finite F p -vector spaces: (3.3) ( I N/K / I K ) / ker( N N/L ) ≃ I L/F / I F (quotient) , I N/K / I K ≃ ( I L/F / I F ) × ker( N N/L ) (direct product) . Figure 2. Hasse subfield diagram of N/F ✉ F = L ∩ K base field ✟✟✟✟✟✟ [ K : F ] = q ✉ K field of degree q [ L : F ] = p ✉ L field of degree p ✟✟✟✟✟✟ ✉ N = L · K compositum of degree p · q Definition 8. Since the relative different of N/K is essentially given by D N/K = Q ti =1 L p − i [33,Thm. 3.2, p. 1993], the space I N/K / I K ≃ Q ti =1 ( h L i i / h L pi i ) of primitive ambiguous ideals of N/K is also called the space of differential factors of N/K . The two subspaces in the direct productdecomposition of I N/K / I K in formula (3.3) are called • subspace I L/F / I F of absolute differential factors of L/F , and • subspace ker( N N/L ) of relative differential factors of N/K .3.2. Splitting off the norm kernel. The second isomorphism in formula (3.3) gives rise to a dichotomic decomposition of the space I N/K / I K of primitive ambiguous ideals of N/K intotwo components, whose dimensions can be given under the following conditions: Theorem 2. Let p ≥ be an odd prime and put q = 2 . Among the prime ideals of L which aretotally ramified over F , denote by p , . . . , p s those which split in N , p i O N = P i P ′ i for ≤ i ≤ s ,and by q , . . . , q n those which remain inert in N , q j O N = Q j for ≤ j ≤ n . Then the space I N/K / I K of primitive ambiguous ideals of N/K is the direct product of the subspace I L/F / I F of absolute differential factors of L/F and the subspace ker( N N/L ) of relative differentialfactors of N/K , whose bases and dimensions over F p are given by (3.4) I L/F / I F ≃ s Y i =1 ( h p i i / h p pi i ) × n Y j =1 ( h q j i / h q pj i ) ≃ F s + np , dim F p ( I L/F / I F ) = s + n, ker( N N/L ) ≃ s Y i =1 (cid:16) h P i ( P ′ i ) p − i / h ( P i ( P ′ i ) p − ) p i (cid:17) ≃ F sp , dim F p (ker( N N/L )) = s. Consequently, the complete space of differential factors has dimension dim F p ( I N/K / I K ) = n + 2 s .Proof. Whereas the qualitative formula (3.3) is valid for any prime p ≥ q > p, q ) = 1, the quantitative description of the norm kernel ker( N N/L ) is only feasible if we put q = 2 and therefore have to select an odd prime p ≥ 3. Replacing N by L and K by F in formula(3.1), we get t = n + s and thus the first isomorphism of formula (3.4). For N and K , however, weobtain t = n + 2 s . We point out that, if s = 0, that is, if none of the totally ramified primes of L/F splits in N , then the induced norm mapping N N/L in formula (3.2) is an isomorphism. Forthe constitution of the norm kernel, see [33, Thm. 3.4 and Cor. 3.3(3), p. 1994]. (cid:3) DANIEL C. MAYER Primitive ambiguous principal ideals. The preceding result concerned primitive ambigu-ous ideals of N/K , which can be interpreted as ideal factors of the relative different D N/K . Formula(3.1) and Theorem 2 show that the F p -dimension of the space I N/K / I K increases indefinitely withthe number t of totally ramified prime ideals of N/K .Now we restrict our attention to the space P N/K / P K of primitive ambiguous principal ideals or differential principal factors (DPF) of N/K . We shall see that fundamental constraints from Galoiscohomology prohibit an infinite growth of its dimension over F p , for quadratic base fields K .3.4. Splitting off the capitulation kernel. We have to cope with a difficulty which arises in thecase of a non-trivial class group Cl K = I K / P K > 1, because then P N/K / P K cannot be viewed asa subgroup of I N/K / I K . Therefore we must separate the capitulation kernel of N/K , that is thekernel of the transfer homomorphism T N/K : Cl K → Cl N , a · P K ( a O N ) · P N , which extendsclasses of K to classes of N :(3.5) ker( T N/K ) = { a · P K | ( ∃ A ∈ N ) a O N = A O N } = ( I K ∩ P N ) / P K . On the one hand, ker( T N/K ) = ( I K ∩ P N ) / P K is a subgroup of I K / P K = Cl K , consisting ofcapitulating ideal classes of K . On the other hand, since I K ≤ I N/K consists of ambiguous idealsof N/K , ker( T N/K ) = ( I K ∩ P N ) / P K is a subgroup of P N/K / P K , consisting of special primitiveambiguous principal ideals of N/K , and we can form the quotient(3.6) ( P N/K / P K ) / (cid:0) ( I K ∩P N ) / P K (cid:1) ≃ P N/K / ( I K ∩P N ) = P N/K / ( I K ∩P N/K ) ≃ ( P N/K ·I K ) / I K . This quotient relation of F p -vector spaces is equivalent to a direct product relation(3.7) P N/K / P K ≃ ( P N/K · I K ) / I K × ker( T N/K ) . Since ( P N/K · I K ) / I K ≤ I N/K / I K is an actual inclusion, the factorization of I N/K / I K in formula(3.3) restricts to a factorization(3.8) ( P N/K · I K ) / I K ≃ ( P L/F / P F ) × (cid:16) ker( N N/L ) ∩ (cid:0) ( P N/K · I K ) / I K (cid:1)(cid:17) , provided that F is a field with trivial class group Cl F , that is I F = P F and thus P L/F / P F ≤I L/F / I F . Combining the formulas (3.7) and (3.8) for the rational base field F = Q , we obtain: Theorem 3. There is a trichotomic decomposition of the space P N/K / P K of differential prin-cipal factors of N/K into three components, (3.9) P N/K / P K ≃ P L/ Q / P Q × (cid:16) ker( N N/L ) ∩ (cid:0) ( P N/K I K ) / I K (cid:1)(cid:17) × ker( T N/K ) , • the absolute principal factors , P L/ Q / P Q , of L/ Q , • the relative principal factors , ker( N N/L ) ∩ (cid:0) ( P N/K I K ) / I K (cid:1) , of N/K , and • the capitulation kernel , ker( T N/K ) , of N/K . Galois cohomology. In order to establish a quantitative version of the qualitative formula(3.9), we suppose that N/K is a cyclic relative extension of odd prime degree p and we use the Galoiscohomology of the unit group U N as a module over the automorphism group G = Gal( N/K ) = h σ i ≃ C p . In fact, we combine a theorem of Iwasawa [20] on the first cohomology H ( G, U N ) witha theorem of Hasse [13] on the Herbrand quotient of U N [18], and we use Dirichlet’s theorem onthe torsion-free unit rank of K . By E N/K = U N ∩ ker( N N/K ) we denote the group of relative units of N/K .(3.10) H ( G, U N ) ≃ E N/K /U σ − N ≃ P N/K / P K (Iwasawa) , ˆH ( G, U N ) ≃ U K /N N/K ( U N ) , with ( U K : N N/K ( U N )) = p U , ≤ U ≤ r + r − θ, ( G, U N ) ( G, U N ) = [ N : K ] = p (Hasse) , where ( r , r ) is the signature of K , and θ = 0 if K contains the p th roots of unity, but θ = 1 else. Corollary 1. If N/K is cyclic of odd prime degree p ≥ , then the F p -dimensions of the spaces ofdifferential principal factors in Theorem 3 are connected by the fundamental equation (3.11) U + 1 = A + R + C, where LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 7 • A := dim F p ( P L/ Q / P Q ) is the dimension of absolute principal factors, • R := dim F p (cid:16) ker( N N/L ) ∩ (cid:0) ( P N/K I K ) / I K (cid:1)(cid:17) is the dimension of relative principal factors, and • C := dim F p (ker( T N/K )) is the dimension of the capitulation kernel. Corollary 2. Under the assumptions p ≥ , q = 2 of Theorem 1, in particular for N dihedral ofdegree p , the dimensions in Corollary 1 are bounded by the following fundamental estimates (3.12) 0 ≤ A ≤ min( n + s, m ) , ≤ R ≤ min( s, m ) , ≤ C ≤ min( ̺ p , m ) , where ̺ p := rank p (Cl K ) , and m := 1 + r + r − θ denotes the cohomological maximum of U + 1 .In particular, we have m = 2 for real quadratic K with ( r , r ) = (2 , or K = Q ( √− if p = 3 , m = 1 for imaginary quadratic K with ( r , r ) = (0 , , except for K = Q ( √− when p = 3 . Remark 2. For N pure metacyclic of degree ( p − p , the space P L/ Q / P Q of absolute principalfactors contains the one-dimensional subspace ∆ = h p √ D i generated by the radicals , and thus(3.13) 1 ≤ A ≤ min( t, m ) , ≤ R ≤ m − , ≤ C ≤ min( ̺ p , m − , where m = p +12 for cyclotomic K with ( r , r ) = (0 , p − ). In particular, there is no capitulation, C = 0, for a regular prime p with ̺ p = 0, for instance p < Remark 3. We mentioned that in general P N/K / P K cannot be viewed as a subgroup of I N/K / I K .In fact, for a dihedral field N which is unramified with conductor f = 1 over K , we have n = s = 0,consequently A = R = 0, and I N/K / I K ≃ P N/K / P K ≃ ker( T N/K ) isat least one-dimensional, according to Hilbert’s Theorem 94 [19], and at most two-dimensional bythe estimate C ≤ min( ̺ p , m ) ≤ min( ̺ p , ≤ §§ N/K .3.6. Differential principal factorization (DPF) types of complex dihedral fields. Let p be an odd prime. We recall the classification theorem for pure cubic fields L = Q ( √ D ) and theirGalois closure N = Q ( ζ , √ D ), that is the metacyclic case p = 3. The coarse classification of N according to the cohomological invariants U and A alone is closely related to the classification of simply real dihedral fields of degree 2 p with any odd prime p by Nicole Moser [36, Dfn. III.1 andProp. III.3, p. 61], as illustrated in Figure 3. The coarse types α and β are completely analogousin both cases. The additional type γ is required for pure cubic fields, because there arises thepossibility that the primitive cube root of unity ζ occurs as relative norm N N/K ( Z ) of a unit Z ∈ U N . Due to the existence of radicals in the pure cubic case, the F p -dimension A of the vectorspace of absolute DPF exceeds the corresponding dimension for simply real dihedral fields by one. Figure 3. Classification of simply real dihedral and pure cubic fields U ✻ A ✲ t α t β U ✻ A ✲ t α t β ❞ γ The fine classification of N according to the invariants U , A , R and C in the simply real dihedralsituation with U + 1 = A + R + C splits type α with A = 0 further in type α with C = 1(capitulation) and type α with R = 1 (relative DPF). In the pure cubic situation, however, nofurther splitting occurs, since C = 0, and R = U + 1 − A is determined uniquely by U and A already. We oppose the two classifications in the following theorems. DANIEL C. MAYER Theorem 4. Each simply real dihedral field N/ Q of absolute degree [ N : Q ] = 2 p with anodd prime p belongs to precisely one of the following differential principal factorization types, independence on the triplet ( A, R, C ) :Type U U + 1 = A + R + C A R Cα α β Proof. Consequence of Cor. 1 and 2. See [36, Dfn. III.1 and Prop. III.3, p. 61] and [26]. (cid:3) Theorem 5. Each pure metacyclic field N = Q ( ζ , √ D ) of absolute degree [ N : Q ] = 6 withcube free radicand D ∈ Z , D ≥ , belongs to precisely one of the following differential principalfactorization types, in dependence on the invariant U and the pair ( A, R ) :Type U U + 1 = A + R A Rα β γ Proof. A part of the proof is due to Barrucand and Cohn [4] who distinguished 4 different types,I ˆ= β , II, III ˆ= α , and IV ˆ= γ . However, Halter-Koch [12] showed the impossibility of one of thesetypes, namely type II. Our new proof with different methods is given in [3, Thm. 2.1, p. 254]. (cid:3) Differential principal factorization (DPF) types of real dihedral fields. Now westate the classification theorem for pure quintic fields L = Q ( √ D ) and their Galois closure N = Q ( ζ , √ D ), that is the metacyclic case p = 5. The coarse classification of N according to the invari-ants U and A alone is closely related to the classification of totally real dihedral fields of degree 2 p with any odd prime p by Nicole Moser [36, Thm. III.5, p. 62], as illustrated in Figure 4. The coarsetypes α , β , γ , δ , ε are completely analogous in both cases. Additional types ζ , η , ϑ are requiredfor pure quintic fields, because there arises the possibility that the primitive fifth root of unity ζ occurs as relative norm N N/K ( Z ) of a unit Z ∈ U N . Due to the existence of radicals in the purequintic case, the F p -dimension A of the vector space of absolute DPF exceeds the correspondingdimension for totally real dihedral fields by one (see Remark 2). Figure 4. Classification of totally real dihedral and pure quintic fields U ✻ A ✲ ✉ α ✉ β ✉ γ ✉ δ ✉ ε U ✻ A ✲ ✉ α ✉ β ✉ γ ✉ δ ✉ ε ❡ ζ ❡ η ❡ ϑ The fine classification of N according to the invariants U , A , R and C in the totally real dihedralsituation with U + 1 = A + R + C splits type α with U = 1, A = 0 further in type α with C = 2(double capitulation), type α with C = R = 1 (mixed capitulation and relative DPF), type α with R = 2 (double relative DPF), type β with U = A = 1 in type β with C = 1 (capitulation), type β with R = 1 (relative DPF), and type δ with U = A = 0 in type δ with C = 1 (capitulation), type δ with R = 1 (relative DPF). In the pure quintic situation with U + 1 = A + I + R [33], however,we arrive at the second of the following theorems where we oppose the two classifications. LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 9 Theorem 6. Each totally real dihedral field N/ Q of absolute degree [ N : Q ] = 2 p with anodd prime p belongs to precisely one of the following differential principal factorization types, independence on the invariant U and the triplet ( A, R, C ) .Type U U + 1 = A + R + C A R Cα α α β β γ δ δ ε Proof. Consequence of the Corollaries 1 and 2. See also [36, Thm. III.5, p. 62] and [26]. (cid:3) Theorem 7. Each pure metacyclic field N = Q ( ζ , √ D ) of absolute degree [ N : Q ] = 20 with -th power free radicand D ∈ Z , D ≥ , belongs to precisely one of the following differentialprincipal factorization types, in dependence on the invariant U and the triplet ( A, I, R ) .Type U U + 1 = A + I + R A I Rα α α β β γ δ δ ε ζ ζ η ϑ The types δ , δ , ε are characterized additionally by ζ N N/K ( U N ) , and the types ζ , ζ , η by ζ ∈ N N/K ( U N ) .Proof. The proof is given in [33, Thm. 6.1]. (cid:3) Remark 4. Our classification of totally real dihedral fields in Theorem 6 refines the classificationby Moser [36] who uses the results on integral representations of the dihedral group D p by Lee [21].She denotes by b = ( U K : N N/K ( U N )) the unit norm index and obtains U N = U K · E N/K as a splitextension (direct product) of U K by E N/K for b = p (types α, β, γ ), and ( U N : U K · E N/K ) = p asa non-split extension (of modules over Z [ D p ]) for b = 1 (types δ, ε ), due to a non-trivial relation N N/K ( H ) = H σ + ... + σ p − = η for the fundamental unit η of K and a unit H ∈ U N \ ( U K · E N/K ).For p = 3, a geometric interpretation of the unit lattice in logarithmic space, i.e., the Dirichlet-Minkowski image of U N , has been given by Hasse [15, 16]. Classifying Angell’s range < d L < 100 000In Table 1 and all the following tables, we present the results of our classification of totally real cubicfields L and their normal closures N into differential principal factorization types τ ( L ) = τ ( N ).The rows correspond to the numerous steps where we applied our algorithm ( § ̺ of the real quadratic subfield K of N and 3-admissible conductors f of N/K . Here, d denotes the fundamental discriminant of K , and q, q , q , resp. ℓ, ℓ , ℓ , denoteprime numbers congruent to 2, resp. 1 modulo 3. In Table 1, the types α and α do not yet occur. Table 1. Totally real cubic discriminants d L = f · d in the range 0 < d L < Multiplicity Differential Principal Factorization f Condition 0 1 2 3 4 α β β γ δ δ ε Total1 ̺ = 0 27089 0 q ≡ d ≡ d ≡ d ≡ d ≡ d ≡ ℓ ≡ q q 30 38 2 38 4 423 q d ≡ q d ≡ q d ≡ q d ≡ q d ≡ qℓ 13 29 1 29 2 313 ℓ d ≡ ℓ d ≡ ℓ d ≡ q q d ≡ ̺ = 1 3300 3300 3300 q ≡ d ≡ d ≡ d ≡ d ≡ d ≡ ℓ ≡ q q q d ≡ q d ≡ q d ≡ ̺ = 2 5 16 4 20Summary 4652 9 21 5 16 10 76 106 3349 79 1117 According to Table 1, the number of non-cyclic totally real cubic fields L with discriminant0 < d L < is , in perfect accordance with the results by Llorente and Oneto [22, 23], whodiscovered the ommission of ten fields in the table by Angell [1, 2]. Together with 51 cyclic cubicfields in Table 2, the total number is (rather than 4794, as announced erroneously in [2]). LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 11 Table 2. Cyclic cubic discriminants d L = f in the range 0 < d L < M DPF f d L f Condition 1 2 ζ d = 1 1 1 9 81 ℓ ≡ +1 (mod 3) 30 30 7 499 ℓ d = 1 4 8 63 3 969 ℓ ℓ ≡ +1 (mod 3) 6 12 91 8 281Summary 31 10 51According to Table 2, the number of cyclic cubic fields L with discriminant 0 < d L < is ,with 31 arising from singlets having conductors f with a single prime divisor, and 20 from doublets having two prime divisors of the conductor f . (M denotes the multiplicity.)Although we have given a succinct survey of the DPF types of all multiplets in Angell’s range0 < d L < in the conclusion of [35], we arrange them again in a more ostensive tabular formwith absolute frequency and minimal discriminant d L = f · d .All doublets in Table 3 are pure . In bigger ranges, there will also occur mixed doublets, e.g. inTable 19. The corresponding 3-class rank is always ̺ = 0. Table 3. Types of doublets in the range 0 < d L < DPF Type Frequency d f d L ( τ ( L ) , τ ( L ))( γ, γ ) 4 33 45 66 825( ε, ε ) 5 373 10 37 300Total: 9The triplets with ̺ = 1 in Table 4 have been partially classified in a coarse sense by Schmithals in1985 [38]. He merely decided whether capitulation occurs or not, indicating C = 1 by the symbol“+” and C = 0 by “ − ”. This admits the detection of type ε but fails to distinguish between thetypes β and δ . Table 4. Types of triplets in the range 0 < d L < DPF Type Frequency d f d L ̺ ( τ ( L ) , . . . , τ ( L ))0 ( γ, γ, γ ) 1 69 18 22 3560 ( ε, ε, ε ) 1 717 9 58 0771 ( β , β , β ) 1 1 509 6 54 3241 ( β , β , ε ) 2 14 397 2 57 5881 ( β , δ , δ ) 3 1 765 7 86 4851 ( δ , δ , δ ) 13 7 053 2 28 212Total: 21The quartets in Table 5 belong to unramified cyclic cubic extensions of quadratic fields with ̺ = 2.In fact, they have been classified by Heider and Schmithals in 1982 [17, p. 24]. In 2006, resp. 2008,resp. 2009, we have detected the remaining capitulation numbers ν ( K ) = 0, resp. 1, resp. 2, whichshow up in Table 16. See [29, 32]. Table 5. Types of quartets in the range 0 < d L < DPF Type Capitulation Number ν ( K ) Frequency d L = d ( τ ( L ) , . . . , τ ( L )) (according to [8])( α , α , α , α ) 4 1 62 501( α , α , α , δ ) 3 4 32 009Total: 54.1. Numerical results by Nicole Moser. In her paper [36] on the units U N and class groupsCl N of dihedral fields N of degree 2 p with an odd prime p , Nicole Moser has given a small table[36, V.4, pp. 72–73] of 34 totally real cubic fields L with discriminants 0 < d L < p = 3. She found 26 fields of type δ ,unramified with conductor f = 1, without exceptions, and thus more precisely of our finer type δ .The frequency ≈ 76% corresponds to Angell’s ≈ d L = 1 · d = d are229 , , , , , , , , , , , , , , , , , , , , , , , , , . In fact, each of the normal closures N of degree 3 over its quadratic subfield K is precisely theHilbert 3-class field F ( K ) of K , with one exception d = , where K has class number 9. Here,Moser’s table entries a = 9, h N = 9 are incorrect and must be replaced by a = 3, h N = 3. She usestwo invariants, the index of subfield units a = ( U N : ( U K · U L · U L σ )) in Formula (9.2), and the unitnorm index b = ( U K : N N/K ( U N )) in Remark 4 for her characterization of the types α, β, γ, δ, ε .Among the remaining 8 fields, one is of type γ with d = 21 ≡ f = 2 · d L = 6 · 21 =756, and seven are of type ε . Among the latter, five have f = 2 and d ∈ { , , , , } , d L ∈ { , , , , } , two have f = 3 and d = 69 ≡ d L = 621, resp. d = 93 ≡ d L = 837. The frequency ≈ 21% corresponds to Angell’s ≈ missing : one is of type γ with d = 13, f = 2 · d L = 10 · 13 = 1300, and three are of type ε with d = 349, f = 2, d L = 1396, resp. d = 57, f = 5, d L = 1425, resp. d = 373, f = 2, d L = 1492.On the other hand, it is very instructive that there is also a superfluous field: although f = 2 is a3-admissible conductor for d = 229, since 2 remains inert in K = Q ( √ d ), D = 2 · d = 4 · 229 = 916is only a formal cubic discriminant, because the defect of 2 is δ (2) = 1. So M d = ∅ is a nilet ,and the given polynomial X − X − X + 4 generates the cubic field with conductor f = 1 anddiscriminant d L = 229. In this case, Moser is uncertain whether the type of the hypothetical cubicfield with discriminant 916 is ε or γ . Type γ , however, is never possible for a field with primeconductor, such as f = 2. Since ̺ = 1 for d = 229, the types β , δ and ε would be possible,but Moser’s claim a = 9 discourages types β and δ . It is mysterious how she determined theinvariant a for a non-existent field without knowing the class number h N . In view of the errors inMoser’s table, it is worth ones while to state a summarizing theorem which also pays attention tothe modest contribution by cyclic cubic fields. Cutting off Table 1 at d L = 1500 we obtain: Theorem 8. Among the totally real cubic fields L with discriminants < d L < , there are ( ) of type γ , ( ) of type δ , and ( ) of type ε . These non-Galois cubic fields arecomplemented by ( ) cyclic cubic fields with conductors f ∈ { , , , , , } . With respectto the multiplicity m , all fields form singlets with m = 1 . Obviously, Moser was not in possession of Angell’s UMT [1], otherwise she would have been ableto detect the gaps in her table. She rather refers to an unpublished table by Ren´e Smadja.Outside of the range 0 < d L < β , moreprecisely our finer type β , for d = 29, f = 2 · d L = 14 · 29 = 5684. The conductor is divisibleby the prime 7 which splits in K , as required for type β .Moser did not know any examples of her type α . From Table 5 we know that the minimaldiscriminant of such a field is 32009, discovered by Heider and Schmithals [17]. Due to f = 1 it ismore precisely our finer type α . See Theorem 14 and Example 11. LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 13 Update of our classification for < d L < 200 000As announced in [35], Table 1 with 0 < d L < was completed on Tuesday, 29 December 2020.One week later, on Tuesday, 05 January 2021, we finished the updated Table 6 containing therevised classification of all totally real cubic fields L with discriminants 0 < d L < · , which wehad investigated in August 1991 [27]. Table 6. Totally real cubic discriminants d L = f · d in the range 0 < d L < · Multiplicity Differential Principal Factorization f Condition 0 1 2 3 4 α α β β γ δ δ ε Total1 ̺ = 0 53848 0 q ≡ d ≡ d ≡ d ≡ d ≡ d ≡ ℓ ≡ q q 65 66 2 66 4 703 q d ≡ q d ≡ q d ≡ q d ≡ q d ≡ qℓ 30 57 3 56 1 6 633 ℓ d ≡ ℓ d ≡ ℓ d ≡ ℓ d ≡ q q q q q d ≡ q q d ≡ q q d ≡ q q ℓ qℓ d ≡ ̺ = 1 6924 6924 6924 q ≡ d ≡ d ≡ d ≡ d ≡ d ≡ ℓ ≡ q q q d ≡ q d ≡ q d ≡ ̺ = 2 16 50 14 64Summary 9702 25 43 16 50 1 21 155 201 7028 188 2301 According to Table 6, the total number of all non-cyclic totally real cubic fields L with discriminants d < · is . Together with 70 cyclic cubic fields in Table 7 the number is 10 015 , Table 7. Cyclic cubic discriminants d L = f in the range 0 < d L < · M DPF f Condition 1 2 ζ d = 1 1 1 ℓ ≡ ℓ d = 1 6 12 ℓ ℓ ≡ cyclic cubic fields L with discriminant 0 < d L < · is ,with 42 arising from singlets having conductors f with a single prime divisor, and 28 from doublets having two prime divisors of the conductor f . (M denotes the multiplicity.)As predicted in the introduction, several fields of type β were unduly classified as type δ ,since Voronoi’s algorithm [40] did not find any absolute principal factors along the chains of latticeminima. Since the table [27, p. 3] accumulates information on conductors with similar behaviorand thus has a format different from Table 6, we compile a translation of critical rows in bothtables, subtracting contributions by cyclic cubic fields in [27]. Table 8. Accumulation of rows in Table 6 for comparison with [27, p. 3]Multiplicity Differential Principal Factorization f Condition 1 2 3 4 α α β β γ δ δ ε Total9 d ≡ ̺ = 0 71 17 48 6 71 ℓ ≡ ̺ = 0 196 45 140 11 196Together 267 62 188 17 267 ℓ ≡ ̺ = 1 5 4 10 1 15Summary 9702 25 43 16 50 1 21 155 201 7028 188 2301 9945[27] m = 1, t = s = 1, ̺ = 0 267 53 198 16 m = 3, t = s = 1, ̺ = 1 5 20 146 f divisible by one prime which splits in the real quadratic field K with3-class rank ̺ = 0, resp. ̺ = 1, that is, m = 1, resp. m = 3, and t = s = 1, n = 0. For ̺ = 0, thereare 10 fields of type δ too much, 9 fields of type β too less, and 1 field of type ε too less. Theyconcern [27, Part I, ̺ = 0, Section 3.1–5 and 3.14] and are corrected in Table 9. For ̺ = 1, thereis 1 field of type δ too much, which is of correct type β . Consequently, there is a correspondingerroneous impact on the row “total”. Wrong counters in [27] are printed with boldface font.We also give arithmetical invariants of the single erroneous triplet explicitly: For d = 568 ≡ f = ℓ = 13 ≡ d L = f · d = 95 992, we have the pure type ( δ , δ , δ ) in[27, Part III, ̺ = 1, Section 2.2] instead of the correct type ( β , δ , δ ), since the Voronoi algorithmdid not find an absolute principal factor and we concluded A = 0 instead of the correct A = 1.In the coarse classification of Schmithals [38], there is no difference between these two types, sinceboth are characterized by (+ , + , +). LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 15 Table 9. Corrections in the range 0 < d L < · No. d L f Erroneous Type Correct Type[27] τ ( L )1 96 481 7 δ β β ε ε β δ β β ε δ β δ β δ β δ β 10 193 857 19 δ β 11 93 217 31 δ β 12 134 540 31 δ β 13 114 005 151 δ β Gain of arithmetical structure for 100 000 < d L < 200 000 . Over real quadratic fields K with 3-class rank ̺ = 0, this range brings some most remarkable gains :(1) first doublet for f = 3 q , d ≡ ε, ε ) ( d = 5 277, q = 2, d L = 189 972),(2) first type γ for f = qℓ ( d = 21, q = 2, ℓ = 43, f = 86, d L = 155 316, singlet ( γ )),(3) first nilet for f = 9 ℓ , d ≡ d = 29, ℓ = 7, f = 63, D = 115 101),(4) first occurrence of a conductor divisible by two splitting primes, s = 2, namely f = 9 ℓ , d ≡ α which requires s ≥ first verification of the Scholz conjecture with unexpected two-dimensional relative principal factorization instead of two-dimensional capitulation( d = 37, ℓ = 7, f = 63, d L = 146 853, singlet ( α ), see Theorems 10 and 20),(5) first occurrence of f = q q q as a doublet of type ( γ, γ ) ( d = 13, f = 110, d L = 157 300),(6) first occurrence of f = 3 q q , d ≡ γ, γ )( d = 213, q = 2, q = 5, f = 30, d L = 191 700),(7) first occurrence of f = 9 q q , d ≡ γ )( d = 13, q = 2, q = 5, f = 90, d L = 105 300),(8) first occurrence of f = q q ℓ as a doublet of type ( γ, γ ) ( d = 37, f = 70, d L = 181 300),(9) first occurrence of f = 3 qℓ , d ≡ γ, γ )( d = 93, q = 2, ℓ = 7, f = 42, d L = 164 052).The following phenomena arise within 3-ring class fields K f , f > 1, over real quadratic fields K with 3-class rank ̺ = 1:(1) first occurrence of f = 3, d ≡ β , δ , ε ( d = 12 081, d L = 108 729, type ( δ , δ , δ ), and d = 19 749, d L = 177 741, type ( β , β , ε )),(2) first occurrence of f = 9, d ≡ δ , δ , δ ) ( d = 1 901, d L = 153 981),(3) first type ε in a triplet with f = ℓ ( d = 3 873, ℓ = 7, d L = 189 777, type ( δ , δ , ε )). Example 1. Hetero geneous multiplets arise from d = 37 with ̺ = 0 and 3-Selmer space V = h η i ,where η denotes the fundamental unit of K = Q ( √ K ) =[ ∅ ; ∅ , ∅ ; ( α )], corresponding to the divisors (1; 7 , 9; 63) of f = 63, since V (7) = V (9) = 0. For (8),we have a quartet Inv( K ) = [ ∅ ; ( ε ) , ∅ , ∅ ; ∅ , ( β ) , ∅ ; ( γ, γ )], corresponding to (1; 2 , , 7; 10 , , 14; 70),the divisors of f = 70, since V (2) = V but V (5) = V (7) = 0. Recall that 148 = 2 · 37 is thewell-known minimum of all non-cyclic positive cubic discriminants. Its type is the singlet ( ε ). Thetype of 45 325 = 35 · 37 is the singlet ( β ). Padding nilets ∅ illuminate the arithmetical structure. Classifying Ennola and Turunen’s range < d L < 500 000The increasing contributions by new types of conductors f in the range 0 < d L < · enforce asplitting into Table 10 for ̺ = 0 and Table 11 for ̺ ∈ { , } . Table 10. Totally real cubic discriminants d L = f · d in the range 0 < d L < · Multiplicity Differential Principal Factorization f Condition 0 1 2 3 4 α α β β γ δ δ ε Total1 ̺ = 0 133534 0 q ≡ d ≡ d ≡ d ≡ d ≡ d ≡ ℓ ≡ q q 147 159 10 161 18 1793 q d ≡ q d ≡ q d ≡ q d ≡ q d ≡ qℓ 83 129 6 125 4 12 1413 ℓ d ≡ ℓ d ≡ ℓ d ≡ ℓ d ≡ ℓ d ≡ ℓ ℓ q q q q q d ≡ q q d ≡ q q d ≡ q q d ≡ q q ℓ qℓ d ≡ qℓ d ≡ qℓ d ≡ non-cyclic totally real cubic fields L with discrim-inants d < · is 26 330 . Together with 110 cyclic cubic fields in Table 12 the number is 26 440 , LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 17 Table 11. Table 10 with 0 < d L < · continued for ̺ ≥ f Condition 0 1 2 3 4 α α β β γ δ δ ε Total1 ̺ = 1 18378 18378 18378 q ≡ d ≡ d ≡ d ≡ d ≡ d ≡ ℓ ≡ q q 15 3 9 93 q d ≡ q d ≡ q d ≡ q d ≡ q d ≡ qℓ 10 1 3 33 ℓ d ≡ ℓ d ≡ ̺ = 2 61 175 69 244Subtotal 18378 127 61 175 89 18714 25 19003 Total According to Table 12, the number of cyclic cubic fields L with discriminant 0 < d L < · is , with 60 arising from singlets having conductors f with a single prime divisor, and 50 from doublets having two prime divisors of the conductor f . (M denotes the multiplicity.) Table 12. Cyclic cubic discriminants d L = f in the range 0 < d L < · M DPF f Condition 1 2 ζ d = 1 1 1 ℓ ≡ ℓ d = 1 9 18 ℓ ℓ ≡ Example 2. In Table 10, the second line with conductor f = q , a prime number q ≡ nilets , starting with the formal cubic discriminant f · d = 2 · not belongto an actual cubic field, and 4296 singlets with minimal discriminant d L = f · d = 2 · 37 = 148 ofan actual cubic field L . Theoretical justifications for these facts are given in [35, Thm. 4.1]: the3- Selmer space V = h η i of the real quadratic field K = Q ( √ d ) is generated by the fundamentalunit η ∈ U K . In the case of a nilet, the 3- ring space mod q , V ( q ), is the null space of codimension δ ( q ) = 1 in V , since η / ∈ O q . In the case of a singlet, we have V ( q ) = V with defect δ ( q ) = 0. Since minor counting errors have occurred in the tables by Moser, Angell and Llorente/Quer(whereas the table by Ennola/Turunen was correct), we explicitly state the ultimate counters oftotally real cubic fields L in five ranges of discriminants 0 < d L < B with various upper bounds B . Theorem 9. The number of cyclic, resp. non-Galois, resp. all, non-isomorphic totally real cubicfields L with discriminants in the range < d L < B is given by (1) 6 , resp. , resp. , for B = 1 500 , (2) 51 , resp. , resp. , for B = 100 000 , (3) 70 , resp. , resp. 10 015 , for B = 200 000 , (4) 110 , resp. 26 330 , resp. 26 440 , for B = 500 000 , (5) 501 , resp. 592 421 , resp. 592 922 , for B = 10 000 000 .Proof. See the tables in sections §§ (cid:3) Recall that no examples of the types α and α occurred in Angell’s range 0 < d L < , and type α remained unknown even in Ennola and Turunen’s range 0 < d L < · . Since this problem isintimately connected with the Scholz Conjecture in § 9, we now emphasize the following theorem. Theorem 10. The minimal discriminants d L = f · d of totally real cubic fields L with conductor f and quadratic fundamental discriminant d such that τ ( L ) is one of the extremely rare differentialprincipal factorization types α , resp. α , are given by (1) 146 853 with f = 63 = 9 · , s = 2 , and d = 37 , ̺ = 0 (unique field in a singlet, m = 1 ),resp. (2) 966 397 with f = 19 , s = 1 , and d = 2 677 , ̺ = 1 (two of the fields in a triplet, m = 3 ).Proof. The unique field L with discriminant 146 853 has been discovered in August 1991 already[27, Part I, ̺ = 0, Section 6.1] and was confirmed in the row with conductor f = 9 ℓ , d ≡ α .The triplet ( L , L , L ) with discriminant 966 397 was found by direct search on 19 November2017. It is now confirmed by gapless construction in the row with conductor f = ℓ ≡ ̺ = 1 in Table 14. According to Theorem 21, the DPF type of the triplet is ( α , α , δ ). (cid:3) Gain of arithmetical structure for 200 000 < d L < 500 000 . The following new featuresarise within 3-ring class fields K f , f > 1, over real quadratic fields K with 3-class rank ̺ = 0,(1) first doublet of type ( ε, ε ) for f = 9 q , d ≡ d = 1 157, q = 2, d L = 374 868),(2) first doublet of type ( ε, ε ) for f = 9 q , d ≡ d = 877, q = 2, d L = 284 148),(3) first doublet of type ( ε, ε ) for f = 3 ℓ , d ≡ d = 597, ℓ = 7, d L = 263 277),(4) first doublet of type ( ε, ε ) for f = 3 ℓ , d ≡ d = 1 068, ℓ = 7, d L = 470 988),(5) first occurrence of f = 9 ℓ , d ≡ β , β ) ( d = 60, ℓ = 7, d L = 238 140),(6) first occurrence of f = ℓ ℓ with s = 2 as singlets of type ( α )( d = 29, f = 91, d L = 240 149 and d = 8, f = 217, d L = 376 712),(7) first singlet of type ( γ ) for f = 3 q q , d ≡ d = 357, f = 30, d L = 321 300),(8) first occurrence of f = 9 q q , d ≡ γ ) ( d = 53, f = 90, d L = 429 300),(9) first singlet of type ( β ) for f = q q ℓ ( d = 93, f = 70, d L = 455 700),(10) first singlet of type ( γ ) for f = 3 qℓ , d ≡ d = 165, f = 42, d L = 291 060),(11) first occurrence of f = 3 qℓ , d ≡ γ, γ ) ( f = 42, d L = 248 724),(12) first occurrence of f = 9 qℓ , d ≡ γ, γ ) ( d = 29, d L = 460 404).The following phenomena arise within 3-ring class fields K f , f > 1, over K with ̺ = 1:(1) first triplet of type ( β , β , ε ) for f = 3, d ≡ d = 52 197, d L = 469 773),(2) first occurrence of f = 9, d ≡ β , β , ε ) ( d = 5 073, d L = 410 913),(3) first occurrence of f = 9, d ≡ β , δ , δ ) ( d = 2 917, d L = 236 277),(4) first occurrence of f = q q , as triplet ( β , β , β ) ( d = 3 173, f = 10, d L = 317 300),(5) first occurrence of f = 3 q , d ≡ β , β , β ) ( d = 5 637, f = 6, d L = 202 932),(6) first occurrence of f = 9 q , d ≡ β , β , β ) ( d = 1 373, f = 18, d L = 444 852),(7) first occurrence of f = qℓ , as triplet ( β , β , β ) ( d = 1 101, f = 14, d L = 215 796).First unramified quartet of type ( δ , δ , δ , δ ) for ̺ = 2 ( d = d L = 214 712 [29, 32]). LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 19 Classifying Llorente and Quer’s range < d L < 10 000 000As opposed to the smaller ranges, the extension to Llorente and Quer’s upper bound 10 caused unexpected complications of twokinds. Firstly, for ramified extensions with conductor f = 2 · d ≡ d L = f · d > d > 15 253, ̺ = 0, resp. d L = f · d > d > 13 117, ̺ = 1. Secondly, for unramified extensions with f = 1 and ̺ = 2,at several discriminants d L = d > Table 13. Totally real cubic discriminants d L = f · d in the range 0 < d L < Multiplicity Differential Principal Factorization f Condition 0 1 2 3 4 6 α α α β β γ δ δ ε ̺ = 0 2623325 q ≡ d ≡ d ≡ d ≡ d ≡ d ≡ ℓ ≡ q q q d ≡ q d ≡ q d ≡ q d ≡ q d ≡ qℓ ℓ d ≡ ℓ d ≡ ℓ d ≡ ℓ d ≡ ℓ d ≡ ℓ ℓ 60 86 2 50 38 1 1 q q q q q d ≡ q q d ≡ q q d ≡ q q d ≡ q q d ≡ q q ℓ 13 35 44 20 1033 qℓ d ≡ qℓ d ≡ qℓ d ≡ qℓ d ≡ qℓ d ≡ qℓ ℓ ℓ ℓ d ≡ q q ℓ Table 14. Table 13 with 0 < d L < continued for ̺ ≥ f Condition 0 1 2 3 4 6 α α α β β γ δ δ ε ̺ = 1 413458 413458 q ≡ d ≡ d ≡ d ≡ d ≡ d ≡ ℓ ≡ q q 534 115 278 12 553 q d ≡ q d ≡ q d ≡ q d ≡ q d ≡ qℓ 364 67 6 160 6 2 6 213 ℓ d ≡ ℓ d ≡ ℓ d ≡ ℓ d ≡ ℓ d ≡ ℓ ℓ q q q q q d ≡ q q d ≡ q q d ≡ q q ℓ qℓ d ≡ qℓ d ≡ ̺ = 2 2870 7951 3529 q ≡ d ≡ d ≡ d ≡ d ≡ ℓ ≡ q d ≡ q d ≡ Total non-cyclic totally real cubic fields L with discriminants d < is 592 421 .Together with 501 cyclic cubic fields in Table 15 the number is 592 922 , in perfect accordance with Belabas [5, p. 1231 and Tbl.6.2, p. 1232], one field less than in the table of Llorente and Quer [24] (the unknown needle in a gigantic hay stack). LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 21 We emphasize the difference between the number of discriminants (without multiplicities),559784 + 2231 + 5543 + 2879 + 5 = 570 442 , and the number of pairwise non-isomorphic fields (including multiplicities in a weighted sum),1 · · · · · 592 421 , which is confirmed by adding the contributions to the 9 DPF types, α , α , α , β , β , γ , δ , δ , ε ,7951 + 142 + 122 + 3924 + 7639 + 9420 + 426972 + 11128 + 125123 = 592 421 . Table 15. Cyclic cubic discriminants d L = f in the range 0 < d L < M DPF f d L f Condition 1 2 4 ζ d = 1 1 1 9 81 ℓ ≡ ℓ d = 1 33 66 63 3 969 ℓ ℓ ≡ ℓ ℓ d = 1 6 24 819 670 761 ℓ ℓ ℓ ≡ cyclic cubic fields L with discriminant 0 < d L < is ,with 217 arising from singlets having conductors f with a single prime divisor, 252 from doublets having two prime divisors of the conductor f , and 32 from quartets having three prime divisors ofthe conductor f . (M denotes the multiplicity.)We point out that cyclic cubic fields are rather contained in ray class fields over Q than in ringclass fields over real quadratic base fields. The single possible DPF type ζ has nothing to do withthe 9 DPF types α , α , α , β , β , γ, δ , δ , ε of non-abelian totally real cubic fields in [34].7.1. Unramified Quartets. According to Theorem 14, the 413 458 unramified singlets N/K withconductor f = 1 over quadratic base fields K with 3-class rank ̺ = ̺ ( K ) = 1 form an overwhelmingcrowd of colorless, monotonous, and boring fields which share the common type δ .In contrast, the N/K over quadratic fields K with ̺ = 2 show aninteresting statistical distribution of types. We consider the type ( τ ( L ) , . . . , τ ( L )) of a quartet( L , . . . , L ) as ordered lexicographically, regardless of permutations. Smallest d see Table 16. Table 16. Types of unramified quartets in the range 0 < d L < DPF Type Capitulation Number ν ( K ) Frequency d L = d ( τ ( L ) , . . . , τ ( L )) (according to [8])( α , α , α , α ) 4 175 62 501( α , α , α , δ ) 3 2391 32 009( α , α , δ , δ ) 2 8 710 652( α , δ , δ , δ ) 1 62 534 824( δ , δ , δ , δ ) 0 234 214 712Total: 2870As known from [29] and [32], the 2391 quartets of mixed type ( α , α , α , δ ) are extremelydominating with a relative frequency of 83 . pure type ( δ , δ , δ , δ ), resp. ( α , α , α , α ). Quartets with mixed type( α , δ , δ , δ ) are rare with 62 hits, and the 8 quartets with mixed type ( α , α , δ , δ ) are almostnegligible. The reason for this behavior is well understood, because the corresponding capitulationtypes κ ( K ) = (ker( T N /K ) , . . . , ker( T N /K )) enforce certain second 3-class groups Gal(F ( K ) /K )of the quadratic base fields K which can be realized easily for the quartets with high frequency,due to modest group orders, but require huge groups in the case of rare quartets (see [30]). Other Multiplets. According to Table 14, the number 2231 of doublets, resp. 5543 of triplets,resp. 2879 of quartets, resp. 5 of sextets, agrees with the corresponding counters given in [24, Tbl.2, p. 588], resp. [24, Tbl. 3, p. 589], resp. [24, Tbl. 4, p. 589], resp. [24, p. 588 and Tbl. 5, p.590], in the paper by Llorente and Quer. However, there are two misprints in the text below Tbl. 4on page 589 of [24], where the authors intended to state that 2870 among the 2879 quartets belongto real quadratic fields K with 3-class rank ̺ = 2, namely the unramified quartets in our Table 16.But the remaining 9 quartets are ramified over real quadratic fields K with 3-class rank ̺ = 0 andshow up in our Table 13. They are analyzed in detail in the following example. Example 3. A common feature of all ( L , . . . , L ) with discriminants in therange 0 < d L < is the congruence class of the quadratic fundamental discriminant d ≡ v ( f ) = 1 and v ( f ) = 2. The reason for theirmultiplicity in terms of 3-defects δ ( f ) (co-dimensions of 3-ring spaces V ( f )) was discussed in [28,Supplements Section, Part 1.a, p. S55, and Part 2.d, pp. S57–S58]. Now we are able to presenttheir differential principal factorizations in Table 17, where the type of the conductor establishesthe connection with Table 13. A generating polynomial for each member L of the quartets is givenin [24, Tbl. 6, p. 591], but we point out that the conductor in the caption of this table should be T = 3 m T > f ), and the discriminant in the table header should be D = 3 m T d (our d L ). Table 17. Nine explicit ramified quartets in the range 0 < d L < No. d L d f Kind of Conductor DPF Type( τ ( L ) , . . . , τ ( L ))1 1 725 300 213 90 = 9 · · q q ( γ, γ, γ, γ )2 2 238 516 141 126 = 9 · · qℓ ( γ, γ, γ, γ )3 2 891 700 357 90 = 9 · · q q ( γ, γ, γ, γ )4 4 641 300 573 90 = 9 · · q q ( γ, γ, γ, γ )5 6 810 804 429 126 = 9 · · qℓ ( γ, γ, γ, γ )6 7 557 300 933 90 = 9 · · q q ( γ, γ, γ, γ )7 7 953 876 501 126 = 9 · · qℓ ( γ, γ, γ, γ )8 8 250 228 4677 42 = 3 · · qℓ ( γ, ε, ε, ε )9 8 723 700 1077 90 = 9 · · q q ( γ, γ, γ, γ ) Example 4. A particular highlight of the range 0 < d L < is the occurrence of ,which did not show up in smaller tables. The reason for their multiplicity in terms of 3-defects δ ( f ) (co-dimensions of 3-ring spaces V ( f ) modulo f in the 3-Selmer space V ) was discussed in[28, Supplements Section, Part 1.c, p. S56, Part 2.b, p. S57, Part 2.d, pp. S57–S58, and Part 2.f, p.S58]. Now we are able to present their differential principal factorizations in Table 18. The leadingtwo sextets are mixed , and the trailing three sextets are pure . The constitution of the sextets isvery heterogeneous: although four of the quadratic fundamental discriminants d admit the irregular contribution 9 to the conductor f only three conductors are actually divisible by 9, but they differeither by the 3-rank ̺ or by the kind of the conductor. A generating polynomial for each member L of the sextets is given in [24, Tbl. 5, p. 590], but again we point out that the discriminant in thetable header should be D = 3 m T d (our d L = f · d ). Types for ̺ = 0 are more simple. Table 18. Five explicit sextets in the range 0 < d L < No. d L d ̺ f Kind of Conductor DPF Type( τ ( L ) , . . . , τ ( L ))1 3 054 132 84 837 ≡ · q ( β , β , δ , δ , δ , ε )2 4 735 467 131 541 ≡ · q ( β , δ , δ , δ , δ , ε )3 5 807 700 717 ≡ · · q q ( γ, γ, γ, γ, γ, γ )4 6 367 572 19 653 ≡ · q ( β , β , β , β , β , β )5 9 796 788 30 237 ≡ · q ( ε, ε, ε, ε, ε, ε ) LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 23 Example 5. We split the 2231 doublets in the range 0 < d L < according to the shape of f . • In Table 19, we begin with two non-split prime divisors of f , that is, we accumulate the resultsfor f = q q , f = 3 q with d ≡ ± f = 9 q with d ≡ ε, ε )is highly dominating over ( γ, ε ) and ( γ, γ ). Here and in the sequel, the given paradigms for d L are not necessarily minimal. Note the constitution 1141 = 429 + 305 + 318 + 89 of the total frequency. Table 19. Types of doublets with two non-split prime divisors of regular f DPF Type Frequency d f d L ( τ ( L ) , τ ( L ))( γ, γ ) 40 33 45 66 825( γ, ε ) 40 9973 10 997 300( ε, ε ) 1061 373 10 37 300Total: 1141 • The irregular situation f = 9 q with d ≡ γ, γ ) is dominating, ( ε, ε ) remains moderate, mixed type ( γ, ε ) is almost negligible. Table 20. Types of doublets with two non-split prime divisors of irregular f DPF Type Frequency d f d L ( τ ( L ) , τ ( L ))( γ, γ ) 245 213 18 69 012( γ, ε ) 6 9213 18 2 985 012( ε, ε ) 89 141 18 45 684Total: 340 • Table 21 reveals that, for f = qℓ , f = 3 ℓ with d ≡ ± f = 9 ℓ with d ≡ f = 9 q with d ≡ ε, ε ) prevails, followed by ( δ , δ ). Table 21. Types of doublets with a split prime divisor of regular f DPF Type Frequency d f d L ( τ ( L ) , τ ( L ))( β , β ) 14 23 717 14 4 648 532( β , δ ) 1 5 061 39 7 697 781( β , ε ) 14 7 589 14 1 487 444( γ, ε ) 9 1 192 65 5 036 200( δ , δ ) 71 4 813 14 943 348( ε, ε ) 327 197 14 38 612Total: 436 • Again, the irregular situation f = 9 ℓ with d ≡ β , β ) dominates over ( ε, ε ). Table 22. Types of doublets with a split prime divisor of irregular f DPF Type Frequency d f d L ( τ ( L ) , τ ( L ))( β , β ) 34 60 63 238 140( ε, ε ) 13 204 63 809 676Total: 47 • In the case of three non-split prime divisors of f , i.e., f = q q q or f = 3 q q with d ≡ ± f = 9 q q with d ≡ γ, γ ), e.g. d = 93, f = 30, d L = 83 700. Example 6. We split the 5543 triplets in the range 0 < d L < according to the shape of f . • Triplets are usually due to elevated 3-class rank ̺ ≥ K . However,the simplest case of triplets with ̺ = 0 arises for the irregular prime power conductor f = 9, d ≡ d = 717, d L = 58 077. Each of the 308 triplets isembedded in a hetero geneous quartet Inv( K ) = [( ε ) , ( ε, ε, ε )]. • There are 77 cases of triplets with irregular conductors f = 9 q , d ≡ ̺ = 0. Theyare all of pure type ( γ, γ, γ ), for instance d = 69, q = 2, f = 18, d L = 22 356. • For the irregular case f = 9 ℓ , d ≡ ̺ = 0, all 14 occurrences are of type ( β , β , β ),for instance d = 177, ℓ = 7, d L = 702 513. There always exists an associated singlet of type ( ε ) withconductor f = 3, that is, the the triplet and the singlet are embedded in a hetero geneous quartetInv( K ℓ ) = [ ∅ , ( ε ) , ∅ , ∅ , ∅ , ( β , β , β )] corresponding to the divisors (1 , , , ℓ, ℓ, ℓ ) of f . • A unique example of f = 9 qℓ , d ≡ ̺ = 0, is given by d = 69, q = 2, ℓ = 13, f = 234, d L = 3 778 164. It is a triplet of mixed type ( β , γ, γ ). • For ̺ = 1 and non-critical split f = ℓ ≡ β , δ , δ ) prevails,followed by mixed type ( δ , δ , ε ). Mixed types have only two distinct components. See Table 23. Table 23. Types of triplets with a non-critical split prime conductor f = ℓ DPF Type Frequency d f d L ( τ ( L ) , τ ( L ) , τ ( L ))( α , α , α ) 10 32 204 7 1 577 996( α , α , δ ) 9 2 677 19 966 397( α , α , δ ) 23 9 749 13 1 647 581( α , δ , δ ) 1 5 477 37 7 498 013( β , β , β ) 4 7 244 19 2 615 084( β , δ , δ ) 226 1 765 7 86 485( δ , δ , δ ) 23 13 688 13 2 313 272( δ , δ , δ ) 1 30 553 13 5 163 457( δ , δ , ε ) 86 3 873 7 189 777( δ , δ , δ ) 2 44 641 7 2 187 409( δ , δ , δ ) 1 54 469 7 2 668 981Total: 386 • For ̺ = 1 and critical split f = 9, d ≡ β , δ , δ ) prevails,followed by mixed type ( δ , δ , ε ). Here, all examples have minimal discriminant d L . See Table 24. Table 24. Types of triplets with critical split prime power conductor f = 9DPF Type Frequency d f d L ( τ ( L ) , τ ( L ) , τ ( L ))( α , α , α ) 4 14 197 9 1 149 957( α , α , δ ) 2 31 069 9 2 516 589( α , α , δ ) 5 15 529 9 1 257 849( α , δ , δ ) 1 30 904 9 2 503 224( β , δ , δ ) 85 2 917 9 236 277( δ , δ , δ ) 6 13 861 9 1 122 741( δ , δ , ε ) 21 15 733 9 1 274 373Total: 124 LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 25 • For ̺ = 1 and non-split f = q ≡ f = 3 or f = 9, d ≡ δ , δ , δ ) is dominating, followed by the mixed type ( β , β , ε ). Mixed type ( β , δ , ε ) with threedistinct components is very rare. See Table 25, where 4088 = 3239 + 359 + 375 + 115. Table 25. Types of triplets with a non-split prime (power) conductorDPF Type Frequency d f d L ( τ ( L ) , τ ( L ) , τ ( L ))( β , β , β ) 304 55 885 2 223 540( β , β , δ ) 160 30 965 2 123 860( β , β , ε ) 640 14 397 2 57 588( β , δ , ε ) 5 417 077 2 1 668 308( β , ε, ε ) 10 492 117 2 1 968 468( δ , δ , δ ) 2869 7 053 2 28 212( δ , δ , ε ) 11 486 461 2 1 945 844( δ , ε, ε ) 35 192 245 2 768 980( ε, ε, ε ) 54 197 445 2 789 780Total: 4088 • Table 26 gives the distribution of DPF types for f = q q , f = 3 q , and f = 9 q , d ≡ β , β , β ) prevails, followed by pure type ( ε, ε, ε ). Table 26. Types of triplets with two non-split ramified primesDPF Type Frequency d f d L ( τ ( L ) , τ ( L ) , τ ( L ))( β , β , β ) 221 3 173 10 317 300( β , β , γ ) 5 63 917 10 6 391 700( β , γ, γ ) 6 82 397 10 8 239 700( γ, γ, γ ) 6 9 413 22 4 555 892( γ, γ, ε ) 3 64 677 10 6 467 700( ε, ε, ε ) 56 9 293 10 929 300Total: 297 • Table 27 shows the triplets with f = qℓ , f = 3 ℓ , and f = 9 q , d ≡ f = 9 ℓ , d ≡ β , β , β ) prevails, followed by pure type ( ε, ε, ε ).Here, 110 = 67 + 7 + 12 + 24. Table 27. Types of triplets with non-split and split ramified primesDPF Type Frequency d f d L ( τ ( L ) , τ ( L ) , τ ( L ))( α , α , α ) 5 6 997 14 1 371 412( β , β , β ) 83 1 101 14 215 796( β , β , β ) 1 21 324 21 9 403 884( β , β , β ) 3 29 317 14 5 746 132( β , β , β ) 1 18 661 18 6 046 164( β , β , γ ) 2 469 62 1 802 836( δ , δ , δ ) 4 24 621 14 4 825 716( ε, ε, ε ) 11 10 733 14 2 103 668Total: 110 • There are only two triplets with two split ramified primes:mixed type ( α , α , β ) for f = ℓ ℓ ( d = 940, f = 91, d L = 7 784 140), andpure type ( α , α , α ) for f = 9 ℓ , d ≡ d = 2 101, f = 63, d L = 8 338 869). We conclude this section on multiplets with information on singlets . Theorem 11. (Ramified and unramified singlets ) (1) A ramified singlet (with conductor f > ) can only be of type ( α ) , ( β ) , ( γ ) , ( δ ) , ( ε ) . (2) An unramified singlet (with conductor f = 1 ) must be of type ( δ ) .Proof. According to the fundamental inequalities in Corollary 2 and the fundamental equation inCorollary 1, we have:(1) For f > 1, the multiplicity formula shows that 3 ̺ divides m [31, Thm. 3.2, p. 2215, Thm.3.3–3.4, p. 2217, and Thm. 4.1–4.2, p. 2224–2225]. Thus, a singlet can only occur for ̺ = 0, and this implies C = 0, i.e. no capitulation can happen. By Theorem 6, we conclude τ ( L ) / ∈ { α , α , β , δ } , and consequently τ ( L ) ∈ { α , β , γ, δ , ε } .(2) For f = 1, we have the multiplicity formula m = (3 ̺ − / m = 1 occurs for ̺ = 1. On the other hand, f = 1 implies t = s = 0, and thus A = R = 0. The fundamental equation degenerates to U + 1 = C , where ̺ = 1 implies the bound C ≤ 1. Thus, C = 1 and U = 0, that is the unique type δ . (cid:3) Example 7. Indeed, singlets of all the types in Theorem 11 actually do occur. Their minimaldiscriminants d L are given in Table 28. Table 28. Smallest occurrences of various singletsDPF Type d f d L ( τ ( L ))( ε ) 37 2 148( δ ) 229 1 229( γ ) 21 6 756( δ ) 53 7 2 597( β ) 29 14 5 684( α ) 37 63 146 853Concerning the frequency of singlets for 0 < d L < , the first row in Table 14 proves thatunramified singlets ( δ ) form the definite hichamp 413 458 among all contributions. The last row(Subtotal) in Table 13 illuminates the second extreme contribution 146 326 by all the other ramifiedsinglets ( α ), ( β ), ( γ ), ( δ ), and clearly dominating ( ε ).Another interesting observation is enabled by the rows with regular conductors f divisible byexactly two primes, i.e. t = 2, in Table 13. It appears that, under certain conditions, non-splitextensions N/K with U K = N N/K ( U N ) in the sense of Remark 4 are forbidden. Generalizing aproof for singlets of type ( γ ) by Moser at the top of p. 74 in [36], we partition the case t = 2,according to the number 0 ≤ s ≤ f which split in the real quadratic field K . The crucial assumption ̺ = 0, that is, the class number of K is not divisible by 3 (andthus capitulation is discouraged, C = 0), implies that there are only three possible types of splitextensions N/K , namely the singlets ( α ), ( β ), ( γ ). Theorem 12. (Ramified singlets of split extensions N/K )Suppose that K is a real quadratic field with -class rank ̺ = 0 , and let f be a regular -admissibleconductor for K with exactly two restrictive prime divisors, t = 2 . Denote by ≤ s ≤ the numberof prime divisors of f which split in the real quadratic field K . Then the following conditions enforcea split extension N/K with U N = U K · E N/K , where E N/K = U N ∩ ker( N N/K ) denotes the subgroupof relative units of N/K . (1) If s = 0 and N has -class number , then N is a singlet of type ( γ ) . (2) If s = 1 and N has -class number , then N is a singlet of type ( β ) . (3) If s = 2 and N has -class number , then N is a singlet of type ( α ) [ or ( γ )] .Proof. The assumption ̺ = 0 implies that the 3-Selmer space V of K is generated by the funda-mental unit η of K . Since we suppose t = 2 with regular restrictive prime divisors q , q of theconductor f , in the sense of Remark 5, η is not contained in the 3-ring spaces V ( q ) and V ( q ),and the multiplicity of f is m = 1, i.e., we have a singlet [31, Thm. 3.3, p. 2217]. LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 27 According to [36, Thm. A, p. 70], the subgroup Cl σN of weakly ambiguous ideal classes of Cl N ,with respect to Gal( N/K ) = h σ i , is of order ( K ) · T − / Q , where the norm index is denotedby 3 Q = ( U K : ( U K ∩ N N/K ( N × ))) and T is the number of prime ideals of K which ramify in N . In our situation, we have ( K ) = 1, and ( N ) = 3 s is divisible by 3 T − − Q , where T = 2 + s , that is, s ≥ s − Q , resp. Q ≥ Q = 1. A fortiori, the unit norm index is b = ( U K : N N/K ( U N )) = ( U K : ( U K ∩ N N/K ( N × ))) · (( U K ∩ N N/K ( N × )) : N N/K ( U N )) = 3, because η is not norm of a number in N × , let alone of a unit in U N . Thus, N/K is a split extension, in thesense of Remark 4, and the types ( δ ) and ( ε ) in item (1) of Theorem 11 are impossible.Equation (9.2) in additive form, V N = 2 · V L + V K + E − 2, where V F := v ( F ) for a numberfield F , degenerates to V N = 2 · V L + E − V K = 0. This gives rise to aparity condition for E :(1) If s = 0 and V N = 0, then 2 · V L = 2 − E and E must be even, E = 2, N of type ( γ ).(2) If s = 1 and V N = 1, then 2 · V L = 3 − E and E must be odd, E = 1, N of type ( β ).(3) If s = 2 and V N = 2, then 2 · V L = 4 − E and E must be even,[either] E = 0, N of type ( α ) [or E = 2, N of type ( γ ), conjectured impossible]. (cid:3) Statistical evaluation and theoretical interpretation of the tables Now we illuminate and analyze our extensive numerical (computational, experimental) results withthe aid of statistical evaluations and theoretical statements.8.1. Stagnation and evolution of arithmetical structures. Some features in the Tables 1, 6,10, 11, 13, and 14 reveal stagnation , that is, multiplicities and DPF types remain constant, andonly the statistical counters show monotonic growth, usually slightly faster than linear. Otherphenomena stick out with conspicuous evolution , leading to new multiplicities and new DPF types.The huge total number 592 922 of all objects occurring in our investigation of the extensive range 0 1) = (3 − 1) = 0.In the following conjectures, of which certain parts are proven theorems, we always give successivepercentages with respect to the upper bounds 10 , 2 · , 5 · and 10 , in this order. Conjecture 1. The relative frequency of unramified nilets with ̺ = 0 slightly decreases from89 . 1% over 88 . 6% and 87 . 9% to 86 . ̺ = 1slightly increases from 10 . 9% over 11 . 4% and 12 . 1% to 13 . δ , showing stagnation . See Theorems 14 and 18. The relative frequency of unramified quartets with ̺ = 2 is marginal below 0 . evolution of types:(1) Up to 10 , = 80% are of mixed type ( α , α , α , δ ), = 20% of pure type ( α , α , α , α ).(2) Up to 2 · , = 87 . 5% are of type ( α , α , α , δ ), = 12 . 5% of type ( α , α , α , α ).(3) Up to 5 · , = 86 . 9% are of mixed type ( α , α , α , δ ), = 6 . 6% of pure type( α , α , α , α ), and also = 6 . 6% of the new pure type ( δ , δ , δ , δ ).(4) Up to 10 , = 83 . 3% are of mixed type ( α , α , α , δ ), = 8 . 6% of pure type( δ , δ , δ , δ ), = 6 . 1% of pure type ( α , α , α , α ), = 2 . 2% of the new mixed type( α , δ , δ , δ ), and = 0 . 3% of another new mixed type ( α , α , δ , δ ). Conjecture 2. For 3-admissible non-split prime(power) conductors f = q , f = 3, and f = 9 with d ≡ K = Q ( √ d ) with 3-class rank ̺ = 0, the relative frequencyof nilets slightly decreases from 73% over 72% and 71% to 69%, and the relative frequency ofsinglets slightly increases from 27% over 28% and 29% to 31%. All singlets are of permanent type ε , showing stagnation . See Theorems 15 and 18. We conjecture the last percentages for the range0 < d L < to be close to their asymptotic limit.8.2. New features for -class rank ̺ = 1 . Since ramified extensions N/K for ̺ = 2 do notoccur in the range 0 < d L < , it is sufficient to state the following theorem for ̺ ≤ Theorem 13. Let K = Q ( √ d ) be a real quadratic base field with fundamental discriminant d and -class rank ̺ ≤ . Suppose f = q · q is a regular -admissible conductor for K with two prime divisors q and q . Then the heterogeneous multiplet M ( K f ) associated with the -ring class field K f mod f of K consists of four homogeneous multiplets M c d , c ∈ { , q , q , f } with multiplicities m (1) , m ( q ) , m ( q ) and m ( f ) . In this order, and in dependence on the -ring spaces V ( q ) , V ( q ) and V ( f ) , these four multiplicities, forming the signature sgn( M ( K f )) of M ( K f ) , are given by (1) (0 , , , , if V ( f ) = V ( q ) = V ( q ) = V ( doublet ), (2) (0 , , , , if V ( f ) = V ( q ) < V ( q ) = V , (3) (0 , , , , if V ( f ) = V ( q ) < V ( q ) = V , (4) (0 , , , , if V ( f ) = V ( q ) = V ( q ) < V ( singlet ),if ̺ = 0 , and thus -Selmer space V is one-dimensional , generated by η ∈ U K = h− , η i , and by (1) (1 , , , , if V ( f ) = V ( q ) = V ( q ) = V ( sextet ), (2) (1 , , , , if < V ( f ) = V ( q ) < V ( q ) = V , (3) (1 , , , , if < V ( f ) = V ( q ) < V ( q ) = V , (4) (1 , , , , if < V ( f ) = V ( q ) = V ( q ) < V ( triplet ), (5) (1 , , , , if V ( f ) < V ( q ) = V ( q ) < V ( nilet with defect δ = 2 ),if ̺ = 1 , and thus -Selmer space V is two-dimensional , generated by η ∈ U K and θ ∈ I \ U K .Proof. These statements are special cases with p = 3 of [35, Thm. 5.1]. (cid:3) Remark 5. We emphasize that in the situation with ̺ = 0 a complete heterogeneous nilet withsignature (0 , , , 0) is impossible, because there always exists a totally real cubic field L withdiscriminant d L equal to either ( q q ) d or q d or q d .This is in contrast to the case ̺ = 1 where a total heterogeneous nilet with signature (1 , , , ramified components, can occur. In this extreme case of a homogeneousnilet M f d with defect δ ( f ) = 2, neither the fundamental unit η nor the other generating 3-virtualunit θ belong to the ring R f modulo f of K , i.e. both of them are deficient .We also point out that Theorem 13 is not only valid for f = q q with primes q i ≡ f = 3 q with q := 3, d ≡ ± q := q ≡ ± f = 9 q with q := 9(the prime power behaves like a prime, formally), d ≡ ± q := q ≡ ± f = q q with any primes q i ≡ ± q i in K , but it is essential that the conductor is regular , that is, 9 ∤ f if d ≡ Example 8. We explicitly consider the statistical results for ̺ = 0, f = q q with q , q ≡ < d L < (Table 13). Since we want to apply probability theory to independent binary properties , we must start with data concerning prime conductors f = q . • Let f = q ≡ q d ,198952 (69%) belong to nilets, realizing the event V ( q ) = 0, and88925 (31%) belong to singlets , realizing the counter event V ( q ) = V . • For f = q q , the four field probability table for independent events yields P = 0 . ≈ . 476 for the event [ V ( q ) = 0 and V ( q ) = 0], P = 0 . · . 31 + 0 . · . ≈ . 214 + 0 . 214 = 0 . 428 for the (symmetric) event[ V ( q ) = 0 and V ( q ) = V ] or [ V ( q ) = V and V ( q ) = 0], P = 0 . ≈ . 096 for the event [ V ( q ) = V and V ( q ) = V ],and these theoretical probabilities are indeed compatible with our experimental result thatamong 6227 admissible discriminants f d ,2706 (43% ≈ . V ( q ) = 0 ∧ V ( q ) = V ] ∨ [ V ( q ) = V ∧ V ( q ) = 0],3092 (50% ≈ . singlets , realizing the event [ V ( q ) = 0 and V ( q ) = 0],429 (7% ≈ . doublets , realizing the event [ V ( q ) = V and V ( q ) = V ].Since almost identical probabilities as for the conductors f = q q with q , q ≡ Conjecture 3. ( Probability for m ∈ { , , } when ̺ = 0)The probabilities P for the occurrence of various multiplets ( L , . . . , L m ) of totally real cubicfields L i among sets of 3-admissible pairs ( f, d ) of regular conductors f and quadratic fundamentaldiscriminants d > ̺ = 0 are approximately given as follows:(1) P ≈ 31% for a singlet, and P ≈ 69% for a nilet, when f = q ,(2) P ≈ 7% for a doublet, P ≈ 50% for a singlet, and P ≈ 43% for a nilet, when f = q q . LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 29 Example 9. Now we present new features for ̺ = 1, f = q q with q , q ≡ < d L < (Table 14). Again, we must begin with prime conductors f = q . • Let f = q ≡ q d ,38302 (92 . V ( q ) < V , and3239 (7 . triplets , realizing the counter event V ( q ) = V . • For f = q q , the four field probability table for independent events yields P = 0 . ≈ . 850 for the event [ V ( q ) < V and V ( q ) < V ], P = 0 . · . 078 + 0 . · . ≈ . 072 + 0 . 072 = 0 . 144 for the (symmetric) event[ V ( q ) < V and V ( q ) = V ] or [ V ( q ) = V and V ( q ) < V ], P = 0 . ≈ . 006 for the event [ V ( q ) = V and V ( q ) = V ],but these theoretical probabilities are not immediately compatible with our experimentalresult that among 649 admissible discriminants f d ,534 (82 . . triplets ,0 (0% ≈ . sextets , realizing the event [ V ( q ) = V and V ( q ) = V ].Only the case of sextets is compatible, in the sense that it has simply not occurred yet in thisrange. At this point, a new phenomenon appears: the possibility of elevated defect δ ( f ) = 2,when 0 = V ( f ) < V ( q ) = V ( q ) < V . We have to split the event [ V ( q ) < V ∧ V ( q ) < V ],with theoretical probability 85 . triplet for 0 < V ( f ) = V ( q ) = V ( q ) < V with experimental probability 17 . nilet for 0 = V ( f ) < V ( q ) = V ( q ) < V with unknown probability, which can now be calculated as 85 . − . 7% =67 . δ = 1 and nilets with δ = 2, that is, 14 . 4% + 67 . 3% = 81 . ≈ . 3% agrees with theexperimental probability for all nilets, indeed. Conjecture 4. ( Probability for m ∈ { , , } when ̺ = 1)The probabilities P for the occurrence of various multiplets ( L , . . . , L m ) of totally real cubicfields L i among sets of 3-admissible pairs ( f, d ) of regular conductors f and quadratic fundamentaldiscriminants d > ̺ = 1 are approximately given as follows:(1) P ≈ 8% for a triplet, and P ≈ 92% for a nilet, when f = q ,(2) P ≈ 1% for a sextet, P ≈ 17% for a triplet, and P ≈ 82% for a nilet, when f = q q .Among the 82% for a nilet, there are 18% nilets with δ = 1 and 82% nilets with δ = 2. Example 10. It is illuminating to give particular realizations of the various multiplets in Theorem13. Let q = 2 and q = 5 and consider the composite conductor f = q q = 10. • Among quadratic fundamental discriminants d with ̺ = 0, there are four d ∈ { , , , } which give rise to nilets M d = ∅ before we find a singlet with conductor 2 for d = 37, d L = 148,and there are eight d ∈ { , , , , , , , } giving rise to nilets M d = ∅ until a singletwith conductor 5 occurs for d = 57, d L = 1425. The consequence of the simultaneous nilets M d = M d = ∅ for d = is the existence of a singlet with conductor f = 10 and d L = 1300in spite of positive defect δ (10) = 1. A nilet M d = ∅ with conductor f = 10 arises for d = 37,because M d is a singlet and M d = ∅ is a nilet. We have to wait for the sixteenth discriminant d for which f = 10 is admissible in order to encounter the first doublet M d for d = 373, d L = 37300with vanishing defect δ (10) = 0. • Among quadratic fundamental discriminants d with ̺ = 1, the probability P ≈ > 69% for anilet with prime conductor is higher, and thus we have to skip 56 discriminants, commencing with d ∈ { , , , . . . } until the first triplet M d with conductor 2 occurs for d = 7053, d L = 28212.Similarly, we must overleap 7 discriminants, beginning with d ∈ { , , , , , . . . } beforewe find a triplet M d with conductor 5 for d = 1257, d L = 31425. Now the new feature of elevateddefect δ = 2 for positive 3-class rank sets in: The consequence of the simultaneous nilets M d = M d = ∅ for d = is not at all a triplet, but rather a nilet M d = ∅ with f = 10, because thering spaces V (2) and V (5) have trivial meet, whence δ (10) = 2. This phenomenon continues forfurther six discriminants starting with d ∈ { , , } until V (10) = V (2) = V (5) coincidefor d = 3173, d L = 317300, giving rise to the first triplet M d . Even later, the first nilet withmoderate defect δ (10) = 1 (it is the 24th in the series of nilets) occurs for d = 7053, since M d is a triplet and M d = ∅ is a nilet. This ostensively shows the dominant role of the 82% nilets M d = ∅ with δ = 2 as opposed to the 18% with δ = 1, according to Conjecture 4. Unramified extensions. The unique conductor without prime divisors is f = 1. It is 3-admissible for any quadratic fundamental discriminant d .Among the 3 039 653 quadratic fundamental discriminants in the range 0 < d < , thereare 2 623 325, resp. 413 458, resp. 2 870, which give rise to real quadratic number fields K = Q ( √ d ) with 3-class rank ̺ = ̺ ( K ) = 0, resp. 1, resp. 2. According to the multiplicity formula m = m ( K, 1) = ̺ − − , there exist 0, resp. 413 458, resp. 11 480, totally real cubic fields L withdiscriminant d L = f · d = 1 · d = d , occurring in nilets , resp. singlets , resp. quartets . Theassociated normal closure N of each of these non-Galois cubic fields L is unramified over its uniquequadratic subfield K . Example 11. The smallest discriminant with ̺ = 0 is d = 5. Although it is an actual quadraticfundamental discriminant, it is only a formal cubic discriminant belonging to a nilet. The minimaldiscriminants d = 229, resp. d = 32 009, with ̺ = 1, resp. ̺ = 2, are both, fundamentaldiscriminants of real quadratic fields and actual discriminants of totally real cubic fields belongingto a singlet, resp. quartet. The latter two discriminants are contained in the table of Angell with0 < d L < already.In the sequel, we briefly speak about the type τ ( N ) = τ ( L ) ∈ { α , α , α , β , β , γ, δ , δ , ε } of atotally real S -field N , resp. its three conjugate cubic subfields L , when we specify the differentialprincipal factorization type of N , resp. L . Theorem 14. Let L be a non-Galois totally real cubic field whose normal closure N is unramifiedover its quadratic subfield K , with conductor f = 1 . (1) If the -class group Cl ( K ) is non-trivial cyclic, then L must be of type τ ( L ) = δ . (2) If K has -class rank ̺ ≥ , then two types τ ( L ) ∈ { α , δ } are possible for L .Proof. See Theorem 18 (1) for item (1), and Theorem 6 with t = s = 0 and thus A = R = 0 foritem (2). (cid:3) Conductors with a single prime divisor.Example 12. It is conspicuous, that the range 0 < d < contains an abundance of 197 nilets with formal cubic discriminants f · d such that the conductor f = q is a prime q ≡ d belongs to a real quadratic field K with 3-class rank ̺ = 2. The smallestvalues of d occurring among these 197 cases are 32 009, 42 817, 62 501. However, the associatedformal cubic discriminants appear in reverse order 250 004 = 2 · 62 501, 1 070 425 = 5 · 42 817,3 873 089 = 11 · 32 009, due to the conductors which increase in the opposite direction. In particular,the smallest formal cubic discriminant 250 004 lies in the range 0 < d < · of Ennola andTurunen already. Actual nonets ( m = 9) of cubic fields with these discriminants do not exist .According to a private communication by Karim Belabas on 31 January 2002, the discriminant 18 251 060 = 2 · f = 2) componentof type α , as required for the proof of the Scholz Conjecture , but even the minimal discriminantof totally real cubic nonets at all (see ). Theorem 15. Let L be a totally real cubic field whose normal closure N is ramified over itsquadratic subfield K with ̺ = 0 and conductor f divisible by a single non-split prime, (1) either f = q a prime q ≡ , inert in K , (2) or f = 3 with d ≡ or d ≡ or f = 9 with d ≡ or f = 9 with d ≡ .In the second and third case, ramifies in K , in the fourth case, remains inert in K .Then L must necessarily be of type τ ( L ) = ε .Proof. See Theorem 18 (2). (cid:3) LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 31 General conditions for differential principal factorizations. The nine possible types τ ( L ) = τ ( N ) ∈ { α , α , α , β , β , γ, δ , δ , ε } of differential principal factorizations of a non-cyclictotally real cubic field L , more precisely of the totally real Galois closure N of L , are defined withthe aid of three invariants A , R and C which are F -dimensions of canonical subspaces of the vectorspace P N/K / P K of primitive ambiguous principal ideals of N over its quadratic subfield K .The most restrictive necessary conditions are imposed by the three types α , α , γ which arecharacterized by two-dimensional subspaces. Theorem 16. (Necessary conditions for two-dimensional subspaces) (1) For type γ with two-dimensional absolute principal factorization A = 2 , the conductor f must have at least two prime divisors, t ≥ . (2) For type α with two-dimensional relative principal factorization R = 2 , the conductor f must have at least two prime divisors which split in K , s ≥ (and a fortiori t ≥ ). (3) For type α with two-dimensional capitulation C = 2 , the -class rank ̺ of K must be atleast two (independently of the conductor f ≥ ).Proof. We make use of the fundamental inequalities in Corollary 2:0 ≤ A ≤ min( n + s, , ≤ R ≤ min( s, , ≤ C ≤ min( ̺, . (1) Type γ ⇐⇒ A = 2 = ⇒ min( n + s, 2) = 2, i.e. t = n + s ≥ α ⇐⇒ R = 2 = ⇒ min( s, 2) = 2, i.e. s ≥ 2, and thus t = n + s ≥ s ≥ α ⇐⇒ C = 2 = ⇒ min( ̺, 2) = 2, i.e. ̺ ≥ (cid:3) Looser necessary conditions are required for non-trivial subspaces. Theorem 17. (Necessary conditions for one-dimensional subspaces) (1) For the types β , β , ε with one-dimensional absolute principal factorization A = 1 , theconductor f must have at least one prime divisor, t ≥ . (2) For the types α , β , δ with one-dimensional relative principal factorization R = 1 , theconductor f must have at least one prime divisor which splits in K , s ≥ (thus t ≥ ). (3) For the types α , β , δ with one-dimensional capitulation C = 1 , the -class rank ̺ of K must be at least one.For each of the types α , β , β , two suitable among these conditions may be combined.Proof. According to the definitions of DPF types and the fundamental inequalities in Corollary 2:(1) Type β , β , ε ⇐⇒ A = 1 = ⇒ min( n + s, ≥ 1, i.e. t = n + s ≥ α , β , δ ⇐⇒ R = 1 = ⇒ min( s, ≥ 1, i.e. s ≥ 1, and thus t = n + s ≥ s ≥ α , β , δ ⇐⇒ C = 1 = ⇒ min( ̺, ≥ 1, i.e. ̺ ≥ (cid:3) Due to the fact that the occurrence of absolute principal factorizations is usually unpredictable assoon as the conductor f > t ≥ 1, there a only very few sufficientconditions for DPF types. Only two types can be enforced unambiguously. Theorem 18. ( Sufficient conditions for types δ and ε ) (1) If N/K is unramified with conductor f = 1 and K has -class rank ̺ = 1 , then τ ( N ) = δ . (2) If the conductor f of N/K has precisely one prime divisor which does not split in K andthe class number of K is not divisible by , then τ ( N ) = ε .In both cases, there exists a unit H ∈ U N such that η = N N/K ( H ) is a fundamental unit of K .Proof. According to the fundamental inequalities in Corollary 2 and the fundamental equation inCorollary 1, we have:(1) t = 0, ̺ = 1 = ⇒ A ≤ min( n + s, 2) = min( t, 2) = 0, s ≤ t = 0, R ≤ min( s, 2) = 0, C ≤ min( ̺, 2) = 1, but on the other hand C = 0 + 0 + C = A + R + C = U + 1 ≥ ⇒ A = R = 0, C = 1 ⇐⇒ Type δ .(2) t = 1, s = 0, ̺ = 0 = ⇒ A ≤ min( n + s, 2) = min( t, 2) = 1, R ≤ min( s, 2) = 0, and C ≤ min( ̺, 2) = 0, but on the other hand A = A + 0 + 0 = A + R + C = U + 1 ≥ ⇒ A = 1, R = C = 0 ⇐⇒ Type ε .In both cases, we obtain U = 0 as a byproduct, i.e. N N/K ( U N ) = U K . (cid:3) Complete verification of the Scholz conjecture Let L be a non-cyclic totally real cubic field. Then L is non-Galois over the rational number field Q with two conjugate fields L ′ and L ′′ . The Galois closure N of L is a totally real dihedral field ofdegree 6, i.e. an S -field, which contains a unique real quadratic field K , as illustrated in Figure 5. Figure 5. Hasse subfield diagram of the normal closure N/ Q of L ✉ Q = L ∩ K rational number field ✟✟✟✟✟✟ [ K : Q ] = 2 ✉ K quadratic field[ L : Q ] = 3 ❡ L L ′ , L ′′ three conjugate cubic fields ✟✟✟✟✟✟ ✉ N = L · K S -field (dihedral field of degree 6) In 1930, Hasse [14] determined the discriminants d L of L [14, pp. 567 (1) and 575] and d N of N [14, p. 566 (2)], in dependence on the discriminant d = d K of K and on the class field theoretic conductor f = f N/K of the cyclic cubic, and thus abelian, relative extension N/K :(9.1) d L = f · d, and d N = f · d . Three years later, Scholz [39, p. 216] determined the relation (9.2) h N = a · h L · h K between the class numbers of the fields N , L and K , in dependence on the index of subfield units , a = ( U N : U ) = 3 E , where U = h U K , U L , U L ′ , U L ′′ i and E ∈ { , , } .Note that E = 0, respectively a = 1, is the distinguished situation where the unit group U N ofthe normal field N is entirely generated by all proper subfield units, that is, U N = U .Scholz was able to give explicit numerical examples [39, p. 216] for E = 1 (e.g. d L = 229), and E = 2 (e.g. d L = 148), but not for E = 0, and he formulated the following hypothesis. Conjecture 5. (The Scholz Conjecture, 1933 , illustrated in Figure 6)There should exist non-Galois totally real cubic fields L whose Galois closure N is either(1) unramified , with conductor f = 1, over some real quadratic field K with 3-class rank ̺ ( K ) = 2 whose complete 3-elementary class group capitulates in N such that U N = U (in the terminology of Scholz, N is an absolute class field over K ) [39, p. 219], or(2) ramified , with conductor f > 1, over some real quadratic field K such that U N = U (here,Scholz calls N a ring class field over K , by abuse of language) [39, p. 221].We point out that, in the unramified situation f = 1, d L = d is a quadratic fundamental discrim-inant, and d N = d is a perfect cube, according to Formula (9.1). In this unramified case, theverification of Conjecture 5 can be obtained from a more general theorem, since any real quadraticfield K with 3-class rank ̺ ( K ) = 2 possesses a multiplet of four unramified cyclic cubic extensions N , . . . , N , that is a quartet of absolutely dihedral fields of degree 6 [29] with non-Galois totallyreal cubic subfields L , . . . , L , each of them selected among three conjugate fields.For such a quartet, Chang and Foote [8] introduced the concept of the capitulation number ≤ ν ( K ) ≤ 4, defined as the number of those members of the quartet in which the complete3-elementary class group of K capitulates. For this number ν ( K ), the following theorem holds. Theorem 19. For each value ≤ ν ≤ , there exists a real quadratic field K with -class rank ̺ ( K ) = 2 and capitulation number ν ( K ) = ν . It is even possible to restrict the claim to fields withelementary -class group of type Cl ( K ) ≃ C × C . LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 33 Figure 6. Hilbert and ring class fields over K ✉ K ❍❍❍❍❍❍ ✟✟✟✟✟✟ [ N : K ] = 3 ✟✟✟✟✟✟ ❍❍❍❍❍❍ unramified quartet ✉ ✉ ✉ ✉ N ✉ (1) ✉ F , ( K ) = F ( K )Hilbert 3-class field of K ✉ K [ N : K ] = 3? ramified singlet ? ❡ N (2) ✉ F ,f ( K ) = K f f of K Proof. From the viewpoint of finite p -group theory, this theorem is a proven statement about thepossible transfer kernel types of finite metabelian 3-groups G with abelianization G/G ′ ≃ (3 , G := Gal( F ( K ) /K ) of K [29]. However, it is easier to giveexplicit numerical paradigms for each value of ν ( K ). We have the following minimal occurrences: ν ( K ) = 4 for d K = 62 501, ν ( K ) = 3 for d K = 32 009, ν ( K ) = 2 for d K = 710 652, ν ( K ) = 1 for d K = 534 824, ν ( K ) = 0 for d K = 214 712,which have been computed by ourselves in [29]. The existence of these cases completes the proof. (cid:3) Remark 6. We have the priority of discovering the first examples of real quadratic fields K with ν ( K ) ∈ { , , } in [29]. However, the first examples of real quadratic fields K with ν ( K ) ∈ { , } are due to Heider and Schmithals [17], who performed a mainframe computation on the CDC Cyberof the University at Cologne, and thus the following corollary is proven since 1982 already. Corollary 3. (Verification of Conjecture 5, (1) for unramified extensions; see Figure 7)There exist non-Galois totally real cubic fields L whose Galois closure N is unramified, with conduc-tor f = 1 , over a real quadratic field K with -class rank ̺ ( K ) = 2 whose complete -elementaryclass group capitulates in N , and which therefore has U N = U . The minimal discriminant of sucha field L is d L = 32 009 (discovered in [17] , actually, the first three members of this quartet withDPF type ( α , α , α , δ ) in Table 5 satisfy the relation U N = U ). Figure 7. Hilbert class field over K ✉ K ❍❍❍❍❍❍ ✟✟✟✟✟✟ [ N : K ] = 3 ✟✟✟✟✟✟ ❍❍❍❍❍❍ unramified quartet ✉ ✉ ✉ ✉ N ✉✉ F , ( K ) = F ( K )Hilbert 3-class field of K Proof. It suffices to take a real quadratic field K with 1 ≤ ν ( K ) ≤ ν ( K ) = 3 and obtain U N = U for d L = d K = 32 009. (cid:3) Concerning the ramified situation f > K . We suppose that he also tacitly assumed a realquadratic field K with 3-class rank ̺ ( K ) = 2. However, more recent extensions of the theory ofdihedral fields by means of differential principal factorizations and Galois cohomology , two conceptswhich we have expanded thoroughly in the preparatory sections §§ U N = U no constraints on the p -class rank ̺ p ( K ) are required. In 1975, Nicole Moser [36] usedthe Galois cohomology ˆH ( G, U N ) ≃ U K / N N/K ( U N ) of the unit group U N of the normal closure N as a module over G = Gal( N/K ) to establish a fine structure with five possible types α, β, γ, δ, ε on the coarse classification by three possible values of the index of subfield units:( U N : U ) = 1 ⇐⇒ type α with ( U K : N N/K ( U N )) = 3,( U N : U ) = 3 ⇐⇒ type β with ( U K : N N/K ( U N )) = 3 or type δ with ( U K : N N/K ( U N )) = 1,( U N : U ) = 9 ⇐⇒ type γ with ( U K : N N/K ( U N )) = 3 or type ε with ( U K : N N/K ( U N )) = 1.Thus, Moser’s refinement does not illuminate the situation U N = U ( ⇐⇒ type α ) of Scholz’sconjecture more closely. Meanwhile, Barrucand and Cohn [4] had coined the concept of (differential)principal factorization ((D)PF) for pure cubic fields. In 1991, we generalized the theory of DPFsfor dihedral fields of both signatures [26], and we obtained a hyperfine structure by splitting Moser’stypes further according to the F p -dimensions C of the capitulation kernel ker( T K,N ) and R of thespace of relative DPFs of N/K , which we recalled in the preparatory section § α with U N = U splits into three subtypes:type α ⇐⇒ C = 2, R = 0, which implies ̺ p ( K ) ≥ α ⇐⇒ C = 1, R = 1, which implies ̺ p ( K ) ≥ f ( s ≥ α ⇐⇒ C = 0, R = 2, which is compatible with any ̺ p ( K ) ≥ 0, but requires s ≥ Conjecture 6. (Conjecture of D. C. Mayer, 1991)Non-Galois totally real cubic fields L whose Galois closure N is ramified , with conductor f > K , and is of type α , with U N = U , should exist for each of thefollowing three situations:(2.1) type α with dim F (ker( T K,N )) = 2 and ̺ ( K ) = 2, s = 0,(2.2) type α with dim F (ker( T K,N )) = 1 and ̺ ( K ) = 1, s = 1,(2.3) type α with dim F (ker( T K,N )) = 0 and ̺ ( K ) = 0, s = 2,where T K,N : Cl ( K ) → Cl ( N ), a · P K ( a O N ) · P N , denotes the transfer homomorphism of3-classes from K to N , and s counts the prime divisors of the conductor f which split in K . Figure 8. Ring class field modulo f = 63 = 3 · K t K [ N : K ] = 3ramified singlet ❞ N t F ,f ( K ) = K f f of K Theorem 20. (Verification of Conjecture 6, (2.3), and Conjecture 5, (2); see Figure 8)There exist non-Galois totally real cubic fields L whose Galois closure N is ramified, with conductor f > divisible by two prime divisors which split in K , i.e. s = 2 , over a real quadratic field K with -class rank ̺ ( K ) = 0 , without capitulation in N , and such that U N = U . The minimaldiscriminant of such a field L is d L = 146 853 = (7 · · (which forms a singlet [27] ).Proof. This was proved in the numerical supplement [27] of our paper [26] by computing a gaplesslist of all 10 015 totally real cubic fields L with discriminants d L < 200 000 on the AMDAHLmainframe of the University of Manitoba. There occurred the minimal discriminant d L = 146 853 = f · d K with d K = 37 and conductor f = 63 = 3 · K , LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 35 i.e. s = 2. This is a necessary requirement for a two-dimensional relative principal factorizationwith R = 2 and is unique up to d L < 200 000. (The next is d L = 240 149 with f = 7 · L with this discriminant d L = 146 853 (forming a singlet). (cid:3) Our discovery of the truth of Theorem 20 with the aid of the list [27] was a random hit withoutexplicit intention to find a verification of Scholz’s conjecture. Unfortunately, [27] does not containexamples of the unique missing DPF type α . It required more than 25 years until we focusedon an attack against this lack of information. In contrast to the techniques of [27], we did notuse the Voronoi algorithm [40] after cumbersome preparation of generating polynomials for totallyreal cubic fields, but rather Fieker’s class field theory routines of Magma [6, 7, 11, 25] for a directgeneration of the fields as subfields of 3-ray class fields modulo conductors f > Theorem 21. (Verification of Conjecture 6, (2.2), and Conjecture 5, (2); see Figure 9)There exist non-Galois totally real cubic fields L whose Galois closure N is ramified, with conductor f > divisible by a single prime divisor that splits in K , i.e. s = 1 , over a real quadratic field K with -class rank ̺ ( K ) = 1 , with one-dimensional capitulation of the elementary -class group in N ,and such that U N = U . The minimal discriminant of such a field L is d L = 966 397 = 19 · (the first two fields of a triplet ( α , α , δ ) , discovered November ).Proof. The proof is conducted in the following section § (cid:3) Figure 9. Heterogeneous quartet modulo f = 19 over K ✉ K ❅❅❅ (cid:0)(cid:0)(cid:0)✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥ heterogeneous quartetunramified singlet ✉ (2.2) ✉ F , ( K ) = F ( K )Hilbert 3-class field of K [ N : K ] = 3 ✟✟✟✟✟✟✟✟✟✟✟✟ ✁✁✁✁✁✁✁✁✁✁✁✁ ❆❆❆❆❆❆❆❆❆❆❆❆ ramified triplet ❡ ❡ ❡ N ✉ F ,f ( K ) = K f f of K Theorem 22. (Verification of Conjecture 6, (2.1), and Conjecture 5, (2); see Figure 10)There exist non-Galois totally real cubic fields L whose Galois closure N is ramified, with conductor f > divisible only by prime divisors which do not split in K , i.e. s = 0 , over a real quadratic field K with -class rank ̺ ( K ) = 2 , with two-dimensional capitulation of the elementary -class groupin N , and such that U N = U . The minimal discriminant of such a field L is d L = 18 251 060 = 2 · (the first five fields of a nonet ( α , α , α , α , α , β , δ , δ , δ ) , discovered November ).Proof. The proof is conducted in the following section § (cid:3) The proof of Theorem 21 and Theorem 22 is conducted in the following sections on real quadraticbase fields with 3-class rank 1 and 2.9.1. Real quadratic base fields with -class rank . In Table 29, we present the results ofour search for the minimal discriminant d L , resp. d N , of a non-Galois totally real cubic field L ,resp. its normal closure N , with differential principal factorization type α . Since ̺ = 1, theunramified component is a singlet , which must be of DPF type δ . Since t = s = 1, the DPF types α , β , β , δ , δ , ε would be possible, for each member of the ramified triplet , but only the types α , δ , δ occur usually. Figure 10. Heterogeneous tridecuplet modulo f = 2 over K ✉ K ❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵ ❅❅❅ (cid:0)(cid:0)(cid:0)✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✟✟✟✟✟✟ ❍❍❍❍❍❍ heterogeneous tridecupletunramified quartet ✉ ✉ ✉ ✉✉ (2.1) ✉ F , ( K ) = F ( K ) Hilbert 3-class field of K [ N : K ] = 3 ✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✜✜✜✜✜✜✜✜✜✜✜✜✜✜✜ ramified nonet ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ N ✉ F ,f ( K ) = K f f of K Table 29. Heterogeneous quartets of S -fields with splitting prime (power) f unramified component ramified components f d d L = f · d δ α δ δ 14 197 1 149 957 1 3 0 07 21 781 1 067 269 1 2 1 013 9 749 1 647 581 1 2 0 119 2 677 966 397 d L = 19 · α . For f = 3 , the condition d ≡ d L = 966 397 for the minimal discriminant, and since the smallestquadratic fundamental discriminant with ̺ = 1 is d = 229, we only have to investigate prime andcomposite conductors f = q d L d K with s ≥ f ≤ r 966 397229 ≈ √ ≈ . , which are divisible by a split prime, that is, f ∈ { , , , 14 = 2 · , 18 = 2 · , , 21 = 3 · , 26 = 2 · , , 35 = 5 · , , 38 = 2 · , 39 = 3 · , 42 = 2 · · , , 45 = 5 · , 57 = 3 · , , 62 = 2 · , 63 = 7 · } . The result of the investigations is summarized in Table 30, which clearly shows that d L = 966 397 ,for d = 2 677 and splitting prime conductor f = 19 bigger than the conductor f = 1 of unramifiedextensions N/K , is the desired minimal discriminant of a totally real cubic field with ramifiedextension N/K , DPF type α and U N = U . The information has been computed with Fieker’sclass field theoretic routines of Magma [11, 25].9.2. Real quadratic base fields with -class rank . In this situation, the unramified quartet is non-trivial, since two DPF types α and δ are possible. These quartets have been thoroughlystudied in [29], and in Table 31 and 32, we use the corresponding notation for capitulation types . LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 37 Table 30. Heterogeneous quartets of S -fields with conductor f , where s = 1unramified component ramified components f condition d d L = f · d δ α β β δ δ d ≡ 966 397 · · d ≡ · d ≡ · d ≡ · 13 21 557 14 572 532 1 3 0 0 0 05 · · 19 13 765 19 876 660 1 3 0 0 0 03 · d ≡ · d ≡ · · d ≡ · · d ≡ · d ≡ · d ≡ · d ≡ · 31 7 573 29 110 612 1 3 0 0 0 07 · d ≡ · d ≡ minimal discriminant d L , resp. d N , of a non-Galois totally real cubic field L , resp. its normal closure N , with differential principalfactorization type α such that N/K is a ramified extension of a real quadratic field K with 3-classrank ̺ = 2. We tried to fix the minimal possible conductor f > 1, namely f = 2. This experimentwas motivated by the fact that the conductor f enters the expression d L = f · d in its secondpower, whereas the quadratic discriminant d enters linearly. Consequently, the probability to findthe minimum of d L is higher for small f than for small d .The table is ordered by increasing quadratic fundamental discriminants d and gives d L = 2 · d and the Artin pattern ( κ , τ ) of the heterogeneous tridecuplet of cyclic cubic relative extensions N/K consisting of an unramified quartet ( N , , . . . , N , ) with conductor f ′ = 1 and a ramified nonet ( N , , . . . , N , ) with conductor f = 2, grouped by the possible two, resp. four, DPF types α , δ ,resp. α , β , δ , ε . Transfer kernels κ are abbreviated by digits, 0 for two-dimensional and 1 , . . . , ∗ for a trivial kernel. Transfer targets τ areabbreviated by logarithmic abelian type invariants of 3-class groups. Symbolic exponents alwaysdenote iteration.The desired minimum is given by d L = 4 · 18 251 060 with five occurrences oframified extensions with DPF type α . Generally, there is an abundance of ramified extensionswith two-dimensional capitulation kernel: at least three and at most all nine of a nonet.Table 32 shows analogous results for the conductor f = 5, that is, d L = 5 · d . The minimum d L = 25 · · d L = 18 251 060 for the minimal discriminant, and since thesmallest quadratic discriminant with ̺ = 2 is d = 32 009, we only have to investigate prime and Table 31. Artin pattern ( κ , τ ) of heterogeneous multiplets modulo f = 2unramified components ramified components α δ α β δ εd K Type κ τ κ τ κ τ κ τ κ τ κ τ a.3 ∗ (1 ) , (1 ) (21 ) ∗ (1 ) (1 ) (1 ) , (1 ) 234 (21 ) (1 ) (1 ) 12 1 , ∗ (1 ) (1 ) (21 ) ∗ (1 ) (1 ) 23 1 , (1 ) , (1 ) (1 ) , (1 ) (1 ) (1 ) 23 (21 ) 11 003 845 a.3 0 (1 ) (1 ) , (21 ) 12 071 253 a.3 0 (1 ) (1 ) 14 266 853 a.3 0 (1 ) , (1 ) 14 308 421 a.3 ∗ (1 ) (1 ) 234 2 , (21 ) , 14 315 765 a.3 0 (1 ) (1 ) 23 (21 ) 14 395 013 a.3 ∗ (1 ) (1 ) 23 (21 ) 15 131 149 D.10 2414 (21) , (1 ) 16 385 741 a.3 ∗ (1 ) (1 ) ) ∗ Table 32. Artin pattern ( κ , τ ) of heterogeneous multiplets modulo f = 5unramified components ramified components α δ α β δ εd K Type κ τ κ τ κ τ κ τ κ τ κ τ (1 ) (1 ) (21 ) (1 ) (1 ) 12 (21 ) ∗ (1 ) , (1 ) 123 (21 ) (1 ) (1 ) ) (2 ) , (1 ) ∗ , (21) (1 ) ∗ composite conductors f = q d L d K with f ≤ r 18 251 06032 009 ≈ √ . ≈ . , that is, f ∈ { , , , · , , , 10 = 2 · , , , 14 = 2 · , 15 = 3 · , , 18 = 2 · , , 21 = 3 · , 22 = 2 · , } . The result of the investigations is summarized in Table 33, which clearly shows that d L = 18 251 060 ,for d = 4 562 765 and the smallest possible conductor f = 2 bigger than the conductor f = 1 ofunramified extensions N/K , is the desired minimal discriminant of a totally real cubic field withramified extension N/K , DPF type α and U N = U . The information has been computed withFieker’s class field theoretic routines of Magma [11, 25]. LASSIFYING MULTIPLETS OF TOTALLY REAL CUBIC FIELDS 39 Table 33. Heterogeneous tridecuplets of S -fields with conductor f unramified components ramified components f condition d d L = f · d α δ α β δ ε d ≡ d ≡ d ≡ d ≡ · d ≡ · d ≡ · · · d ≡ · d ≡ · d ≡ · d ≡ · d ≡ · d ≡ · d ≡ · 11 2 706 373 1 309 884 532 3 1 6 3 0 09.3. Scholz conjecture for p ≥ . We have been curious if the conjecture of Scholz can alsobe verified for dihedral fields N/ Q of degrees 10 and 14. This is indeed the case, and the rootdiscriminants f · d in the following theorem are probably minimal. Theorem 23. Let p be an odd prime number. Suppose L is a non-Galois number field of degree p with totally real absolutely dihedral Galois closure N of degree p , and let K be the unique realquadratic subfield of N . Then N satisfies the condition U N = U := h U K , U L , U L (1) , . . . , U L ( p − i , (1) if d L = ( f · d ) with f = 11 · , d = 5 , f · d = 581 405 , when p = 5 , (2) if d L = ( f · d ) with f = 29 · , d = 13 , f · d = 20 215 117 , when p = 7 .In both cases, the conductor is of the form f = ℓ · ℓ with prime numbers ℓ i ≡ +1 (mod p ) whichsplit in K , and L is a singlet with differential principal factorization type τ ( L ) = α .Proof. By immediate inspection of real quadratic fields K with p -class rank ̺ p = 0 and p -admissibleconductors f , divisible by two primes which split in K , with the aid of Magma. (cid:3) Conclusion In this paper, we have given the complete classification of all multiplets of totally real cubic fields L in the range 0 < d L < of Llorente and Quer [24] according to their differential principalfactorizations (Tables 13 and 14). Inspired by discussions after our two presentations at the WestCoast Number Theory Conference in Asilomar, December 1990, we had attempted this classificationin August 1991 already, but we were forced to restrict the range to the upper bound 2 · in [27].In spite of the required correction of 14 errors (Tables 8 and 9), the table [27] and the associatedtheory [26] were a masterpiece of outstanding innovations concerning DPF types of multiplets ofdihedral fields and a role model for the present paper and its predecessor [35]. We have also given the complete verification of the Conjecture of Arnold Scholz (Conjecture 5).It was necessary to develop the new concept of relative principal factorizations in order to illuminatethe full reach of this conjecture, which we have reformulated more ostensively in Conjecture 6. Dueto the computational challenges, the proof of each of the different perspectives of the conjecture wasestablished many years after Scholz’s paper in 1933 [39]: Corollary 3 on f = 1 was proved 49 yearslater in 1982 [17], Theorem 20 concerning the type α singulet 58 years later in 1991 [27], Theorem21 on the triplet containing type α even 84 years later on 19 November 2017, and Theorem 22 onthe nonet containing type α with f > asymptotic tendencies of DPF types, basedon five ranges of discriminants 0 < d L < B with increasing upper bounds B . Relative frequenciesare rounded to integer percentages. It is striking that the normal closures N of an overwhelmingproportion with 93% of all totally real cubic fields L have a unit group U N which is a non-splitextension of U K = h− , η i if considered as a module over the integral group algebra Z [ S ], since itcontains a unit H such that N N/K ( H ) = H · H σ · H σ = η , according to Remark 4.This phenomenon is due to extremely dominating unramified extensions N/K with conductor f = 1, ̺ = 1, and mandatory type δ (72%), and ramified extensions N/K with regular prime(power)conductor f , ̺ = 0, and mandatory type ε (21%). In contrast, the contributions by the rare types α and α and by the cyclic cubic fields ζ are in fact negligible . In spite of its distinctive dominancefor conductors with two or more prime divisors, type γ remains marginal with 2%. 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