aa r X i v : . [ m a t h . R A ] M a y NONCROSSED PRODUCTS IN WITT’S THEOREM
TIMO HANKE AND JACK SONN
Abstract.
Since Amitsur’s discovery of noncrossed product di-vision algebras more than 35 years ago, their existence over morefamiliar fields has been an object of investigation. Brussel’s workwas a culmination of this effort, exhibiting noncrossed productsover the rational function field k ( t ) and the Laurent series field k (( t )) over any global field k — the smallest possible centers ofnoncrossed products.Witt’s theorem gives a transparent description of the Brauergroup of k (( t )) as the direct sum of the Brauer group of k andthe character group of the absolute Galois group of k . We clas-sify the Brauer classes over k (( t )) containing noncrossed productsby analyzing the fiber over χ for each character χ in Witt’s the-orem. In this way, a picture of the partition of the Brauer groupinto crossed products/noncrossed products is obtained, which isin principle ruled solely by a relation between index and numberof roots of unity. For large indices the noncrossed products occurwith a “natural density” equal to 1. Introduction
A finite-dimensional division algebra is called a crossed product if oneof its maximal commutative subfields is Galois over the center of thealgebra, otherwise a noncrossed product . The existence of noncrossedproducts was for several decades the biggest open question in the theoryof finite-dimensional division algebras before it was settled by Amitsur[2] in 1972.
Date : November 14, 2018.2000
Mathematics Subject Classification.
Primary 16K20, Secondary 11R32;16S35.
Key words and phrases. noncrossed product, Witt’s theorem, Brauer group, den-sity, Laurent series, function field, global field, Galois cover, full local degree, height,root of unity.The first author was supported by the Technion — Israel Institute of Technologyand by the 5th Framework Programme of the European Commission (HPRN-CT-2002-00287).The second author was supported by the Fund for Promotion of Research at theTechnion.
The present paper is motivated by the work of Brussel [5,6] on theexistence of noncrossed products over the fields k ( t ) and k (( t )) for anyglobal field k , which determined these fields to be the “smallest” pos-sible centers of noncrossed products.Brussel’s result is even more surprising over k (( t )) than over k ( t )because of the much simpler structure of its Brauer group. Due to Witt’s theorem [17], if k is perfect, there is a canonical isomorphismBr( k (( t ))) ∼ = Br( k ) ⊕ X ( G k ) (1.1)(depending on t ), where X ( G k ) = Hom c ( G k , C ∗ ) denotes the group ofcontinuous characters of the absolute Galois group of k . Each element χ of the torsion group X ( G k ) belongs to a finite cyclic extension field k ( χ ) ⊇ k with degree equal to the order | χ | of χ . ( k ( χ ) is the fixedfield of ker χ .) Moreover, a canonical generator σ χ of the Galois groupof k ( χ ) over k is obtained by restricting to k ( χ ) any σ ∈ G k with χ ( σ ) = e πi/ | χ | . Under (1.1), χ maps to the class of the cyclic algebra( k ( χ )(( t )) /k (( t )) , σ χ , t ) over k (( t )).If k is non-perfect then Br( k ) ⊕ X ( G k ) is a subgroup of Br( k (( t ))), theso-called “tame” part. Throughout the paper we will work exclusivelyinside this subgroup, and the global field k may be perfect or not.Let k be a global field, i.e. either a number field or a function fieldin one variable over a finite field. Brussel proves the existence of pairsof α ∈ Br( k ) and χ ∈ X ( G k ) such that the k (( t ))-division algebra rep-resenting α + χ is a noncrossed product. In this way, all indices forwhich noncrossed products are known to exist can be realized. How-ever, it has not yet been determined for all classes α + χ whether therepresenting division algebra is a noncrossed product. This task is themotivation for the present paper.Our approach is to partition Br( k ) ⊕ X ( G k ) into the union of fibersover χ where χ runs through X ( G k ) and to classify the noncrossedproducts in each fiber. (Division algebras are identified with theirrespective Brauer classes.)Let χ ∈ X ( G k ) and consider all division algebras of fixed index N ∈ N inside Br( k ) + χ . We will show that one of the following twocases occurs:(I) All division algebras in the fiber over χ of index N are crossedproducts.(II) Among all division algebras in the fiber over χ of index N thenoncrossed products have a “natural density” equal to 1. Inparticular, there are infinitely many. ONCROSSED PRODUCTS IN WITT’S THEOREM 3
Moreover, for fixed χ and varying N the cases (I) and (II) are separatedby bounds on the prime powers dividing N in such a way that, roughlyspeaking, case (I) occurs “below” the bounds and case (II) “above”.The details are as follows. By a well-known formula (cf. (6.1) below) allelements in Br( k ) + χ have index a multiple of | χ | . Let N = | χ | m andlet Q p n p be the prime factorization of m . There are simply definedbounds b p ( χ ) where p runs through the prime factors of | χ | , each b p ( χ )a nonnegative integer or infinity, such that we are in case(I) if n p ≤ b p ( χ ) for all prime factors p of | χ | ,(II) if n p > b p ( χ ) for some prime factor p of | χ | .Analogous results hold over k ( t ) as well as over k (( t )). Furthermore,the field k (( t )) can be replaced by any discrete rank one Henselianvalued field with residue field k . Acknowledgements.
The first author would like to thank the Depart-ments of Mathematics at Bar-Ilan University and at the Technion fortheir kind hospitality and support during his visits in 2006 and grate-fully acknowledges the financial support from the European Commis-sion and the Technion.2.
Notation and Statement of Results
Let k be a field and k sep its separable closure. We write G k forGal( k sep /k ), the absolute Galois group of k . The nonnegative integer s p ( k ), p prime, is defined by the condition that Q p s p ( k ) is the numberof roots of unity contained in k . (If p = char k then s p ( k ) = 0.)Denote by X ( G k ) the group Hom c ( G k , C ∗ ) of continuous characters.( G k is a pro-finite group with the Krull topology and C ∗ is equippedwith the discrete topology.) X ( G k ) is an abelian torsion group. Foreach χ ∈ X ( G k ), the fixed field of ker χ is a cyclic extension of k ofdegree | χ | that we shall denote by k ( χ ). We abbreviate s p ( k ( χ )) to s p ( χ ).Since X ( G k ) is an abelian torsion group, each of its p -primary com-ponents X p ( G k ) is a direct summand. We let χ p denote the projectionof χ ∈ X ( G k ) onto X p ( G k ). As an element of the abelian p -group X p ( G k ), χ p has a height , which is defined to be the maximal nonnega-tive integer r such that χ p is divisible by p r , or infinity if no maximal r exists (cf. Kaplansky [10], §
9, p.19). We denote the height of χ p byht p ( χ ).In terms of the corresponding cyclic field extensions, k ( χ p ) is theunique subfield of k ( χ ) with maximal p -power degree over k . Moreover,ht p ( χ ) is the maximal nonnegative integer r such that the cyclic exten-sion k ( χ p ) /k embeds into a cyclic extension L/k with [ L : k ( χ p )] = p r , TIMO HANKE AND JACK SONN or infinity if no maximal r exists. Since ht p ( χ ) is an invariant of thefield extension k ( χ ) /k , the notation ht p ( k ( χ ) /k ) is also valid. Definition 2.1.
Let χ ∈ X ( G k ).a) χ is said to be exceptional if k is a number field, i := √− ∈ k ( χ )and ht ( k ( χ ) /k ( i )) > ht ( k ( χ ) /k ) > . b) For each prime p define b p ( χ ) := ( ht p ( χ ) + s p ( χ ) + 1 if p = 2 and χ is exceptional,ht p ( χ ) + s p ( χ ) otherwise.Being exceptional as well as the numbers b p ( χ ) are all invariants ofthe field extension k ( χ ) /k . Therefore, we can speak of an exceptionalextension k ( χ ) /k and we can write b p ( k ( χ ) /k ).Clearly, χ is exceptional if and only if χ is exceptional.The extension k ( i ) /k is exceptional if and only if k is a number field, i k and ∞ > ht ( k ( i ) /k ) >
0. This is the case for all k = Q ( √− a )with a a positive integer ≡ Remark.
It can be shown that exceptional characters exist also with K ) k ( i ) and with s ( χ ) taking all values ≥
2. Details will appear ina separate paper [9].
Theorem 2.2 (Main Theorem) . Let k be a global field and let χ ∈ X ( G k ) be a character. The division algebras in the fiber Br( k ) + χ allhave index | χ | m for some positive integer m . Let m be fixed and let m = Q p n p be its prime factorization.(i) If n p ≤ b p ( χ ) for all prime factors p of | χ | then all divisionalgebras in the fiber over χ of index | χ | m are crossed products.(ii) If | χ | has a prime divisor p with n p > b p ( χ ) then the fiber over χ contains noncrossed product division algebras of index | χ | m . More-over, among all division algebras in the fiber over χ of index | χ | m thenoncrossed products have a “natural density” equal to . For algebras of p -power index ( p prime) the picture becomes par-ticularly simple: if | χ | and m are both powers of p then we are incase (I) if n p ≤ b p ( χ ), (II) if n p > b p ( χ ).Since it is known that ht p ( χ ) = ∞ for p = char k , there are no non-crossed product p -algebras in Br( k ) ⊕ X ( G k ).The proof of Theorem 2.2 for exceptional characters χ is more diffi-cult than that for non-exceptional χ , so the additional pieces requiredfor the exceptional characters are separated out and presented later, in ONCROSSED PRODUCTS IN WITT’S THEOREM 5 the interest of greater readability, after the proof for non-exceptionalcharacters is complete. On the other hand, there are interesting as-pects to this part of the proof of the exceptional case, involving non-routine applications of the special case of the local-global principle forthe height of a cyclic extension of a number field, and of Neukirch’stheorems on embedding problems with prescribed local solutions. Theembedding problems arise for a family of metacyclic 2-groups whichare exceptional in their own way (in the classification of metacyclic p -groups). 3. Preliminaries
Cyclotomic fields.
Let k be any field. We denote by µ ( k ) , µ n ( k )and µ p ∞ ( k ) the group of all roots (resp. all n -th roots, all p -powerroots) of unity contained in k . Given that k is fixed, µ, µ n , µ p ∞ are therespective groups of roots of unity contained in k sep . Hence, k ( µ n ) isthe n -th cyclotomic field over k .We choose once and for all a system of primitive roots of unity ζ m such that ζ nm = ζ m/n for all n | m . We set η m := ζ m + ζ − m and i := √− k ( η m ) do not depend on the choice of ζ m .For fields of characteristic zero we define ˜ k := k ∩ Q ( µ ∞ ). Recallthat (3.1) below is the full subfield lattice of Q ( µ ∞ ) in which all linesindicate quadratic extensions. Q ( ζ s +1 ) Q ( η s +1 ) Q ( iη s +1 ) Q ( ζ s ) Q ( η s ) Q ( ζ ) Q ( η ) Q ( iη ) Q ( i ) Q ♣♣♣♣♣♣ ✻ ♣♣♣♣♣♣♣ ✻ ✏✏✏✏✏✏✏✏✑✑✑ ♣♣♣♣ ❄✑✑✑✏✏✏✏✏✏✏✏✏ ♣♣♣♣ ❄ ✏✏✏✏✏✏✏✏✏✑✑✑✑✑✑✑✏✏✏✏✏✏✏✏✏✏ (3.1) Local fields.
By a local field we mean a finite extension of Q p or F p (( t )). Let K/k be a Galois extension of local fields. We write k for TIMO HANKE AND JACK SONN the residue field. Let I denote the maximal unramified subextensionof K/k , q the number of elements in the residue field of k , and e theramification index of K/k . We suppose
K/k is tamely ramified, i.e.char k ∤ e . Then K/I is a cyclic Kummer extension of exponent e (cf. Lang [11, Chapter II, §
5, Proposition 12, p. 52]). In particular, µ e ⊂ I and Gal( K/k ) is metacyclic. Due to Kummer theory, theaction of Gal(
I/k ) on Gal(
K/I ) is given by the action of Gal(
I/k ) on µ e . The latter lifts from the residue field, so the Frobenius element ofGal( I/k ) acts on Gal(
K/I ) as the q -th power map. Thus, Gal( K/k )has a presentation h x, y | x e = 1 , y f = x t , x y = x q i (3.2)for some t | e . Here, y | I is the Frobenius and x is a certain genera-tor of Gal( K/I ). (Note that x has to be suitably chosen in orderto achieve t | e .) The parameters are further subject to the relations q f ≡ e ) and q ≡ e/t ) (because y f commutes with x and x t commutes with y respectively). Note that K/k is abelian ⇐⇒ q ≡ e ), (3.3)in which case K/k is cyclic if and only if t = 1. Global fields.
By a global field we mean a finite extension of Q or afinite extension of F p ( t ). Let k be a global field. For any prime p of k — archimedian or non-archimedian — we write k p for the completionof k at p . If F/k is Galois then F p denotes the completion of F at anyprime of F dividing p . ( F p is unique up to k -isomorphism.)Now, let p be a non-archimedian prime of k . We denote by N ( p )the absolute norm of p , i.e. the number of elements in the residue fieldof p . Let F/k be a finite Galois extension and let P be a prime of F dividing p . We denote by Z P ( F/k ) , I P ( F/k ) , e p ( F/k ) and ϕ P ( F/k )respectively the decomposition field, the inertia field, the ramificationindex and the Frobenius element for P ( e p does not depend on thechoice of P , for F/k is Galois). Recall that Gal( I P /Z P ) is generatedby ϕ P . One says p splits completely in F if Z P ( F/k ) = F for all P | p ,and p is unramified in F if I P ( F/k ) = F for all P | p . For instance, in k ( µ m ) any p with ( N ( p ) , m ) = 1 is unramified and p splits completely in k ( µ m ) ⇐⇒ N ( p ) ≡ m ) , (3.4) p splits completely in k ( η m ) ⇐⇒ N ( p ) ≡ ± m ) . (3.5) ONCROSSED PRODUCTS IN WITT’S THEOREM 7
Chebotarev’s Density Theorem (cf. Weil [16], Chapter XIII, § σ ∈ Gal(
F/k ) is the Frobe-nius element for infinitely many primes P of F . Applied to a composi-tum F F of two Galois extensions F , F ⊇ k this means that for any σ ∈ Gal( F /k ) and σ ∈ Gal( F /k ) coinciding on F ∩ F , there areinfinitely many non-archimedian primes P of F F such that σ i is theFrobenius element for P ∩ F i , i = 1 , The Height
Let
K/k be a cyclic field extension. In the study of ht p ( K/k ) weassume [ K : k ] to be a p -power, for ht p ( K/k ) depends by definitiononly on the maximal p -power subextension of K/k . However, all resultsof this section formally hold for arbitrary [ K : k ]. Before focusing onlocal and global fields we mention two important facts over arbitraryfields. The first one is known as Albert’s theorem. If µ p r ⊆ k then ht p ( K/k ) ≥ r ⇐⇒ µ p r ⊂ N K/k ( K ∗ ) . Proof.
Albert [1, Chapter 9, Theorem 11]. (cid:3)
For instance, − Q ( √− / Q , so ht ( Q ( √− / Q ) =0. The second fact is Artin-Schreier theory. If char k = p and p | [ K : k ] then ht p ( K/k ) = ∞ . Proof.
Albert [1, p. 194f]. (cid:3)
For local and global fields we have ht p ( k/k ) = ∞ . Hence, ht p ( K/k ) = ∞ for all p not dividing [ K : k ]. But infinite height is also possible innon-trivial cases and in characteristic zero: For instance, Q ⊂ Q ( √
2) = Q ( η ) ⊂ Q ( η ) ⊂ . . . is an infinite tower of cyclic number fields of 2-power degree, hence ht ( Q ( η i ) / Q ) = ∞ for all i . Proposition 4.1.
Suppose
K/k is a cyclic extension of local fields.Let e be the ramification index of K/k and let v p ( e ) denote the p -adicexponential value of e . Then ht p ( K/k ) = ( ∞ if p ∤ e , s p ( k ) − v p ( e ) if p | e and p = char k .Proof. Assume [ K : k ] is a p -power. If p ∤ e then K/k is unramified,so ht p ( K/k ) = ∞ . Assume now p | e and p = char k . Let I denote themaximal unramified subfield of K . Write s = s p ( k ) and v = v p ( e ). TIMO HANKE AND JACK SONN
The inequality ht p ( K/k ) ≥ s − v follows from Albert’s Theorem.Indeed, since ζ p s is a norm in I/k and [ K : I ] = e = p v , ζ p s − v is a normin K/k . Conversely, let
L/k be a cyclic extension containing K with[ L : K ] = p h . We show h ≤ s − v . The intermediate fields of L/k are linearly ordered. Therefore, since I ( K , the maximal unramifiedsubfield of L/k equals I . Hence, the ramification index of K/k is p v + h .By (3.3), v + h ≤ s . (cid:3) Lemma 4.2.
Suppose
K/k is a cyclic extension of global fields. Then ht p ( K/k ) ≤ ht p ( K p /k p ) for any prime p in k .Proof. We may assume [ K : k ] is a p -power. Suppose L/k is cyclic, L ⊇ K and [ L : K ] = p r . Let p be a prime in k and claim ht p ( K p /k p ) ≥ r . Let Z be the (unique) decomposition field of p in L/k . Since thesubfields of
L/k are linearly ordered we have Z ⊇ K or Z ⊆ K . Inthe former case K p = k p , hence ht p ( K p /k p ) = ∞ . In the latter case[ L p : K p ] = [ L : K ], hence ht p ( K p /k p ) ≥ r . (cid:3) Example 4.3. If p, q are primes with q = ap n + 1 and ( a, p ) = 1 thenthe degree p m subfield K of the q -th cyclotomic field Q ( µ q ), 1 < m ≤ n ,has ht p ( K/ Q ) = n − m . Proof.
Since Q ( µ q ) is cyclic of degree q −
1, it contains a cyclic extensionof degree p n . Thus, ht p ( K/ Q ) ≥ n − m . Conversely, the prime q istotally ramified in Q ( µ q ), in particular also in K . Hence, by Lemma4.2 and Proposition 4.1, ht p ( K/ Q ) ≤ ht p ( K q / Q q ) = s p ( Q q ) − v p ( p m ) = n − m. (cid:3) If the global field k and a prime p = char k are fixed then a gen-eral (i.e. random) cyclic extension K/k of degree divisible by p hasht p ( K/k ) = 0. This is the interpretation of
Proposition 4.4.
Let k be a global field. Then X p ( G k ) /X p ( G k ) p isinfinite for any p = char k . (Note that X p ( G k ) p is the subgroup consisting of characters of heightgreater than zero.) Proof.
We write X p ( G k ) additively (only in this proof) and show that X p ( G k ) /pX p ( G k ) is infinite. Set t := s p ( k ( µ p )). By Chebotarev’s den-sity theorem there are infinitely many primes p of k which split com-pletely in k ( µ p ) and do not split completely in k ( µ p t +1 ). Let S = { p , ..., p n } be any finite set of such p . By the Grunwald-Wang Theo-rem [3, Chapter X, Theorem 5], there exists a cyclic extension L i /k ofdegree p t in which p i is totally ramified, and in which p j splits com-pletely for every j = i . Let χ i be a character of G k corresponding to ONCROSSED PRODUCTS IN WITT’S THEOREM 9 L i for i = 1 , ..., n . Let χ = P a i χ i with some a i , say a , not divisibleby p . Then if L is the cyclic extension corresponding to χ , the comple-tion of L at p coincides with the completion of L at p , which doesnot embed into a cyclic extension of degree p t +1 of k p . Hence L/k likewise does not embed into a cyclic extension of degree p t +1 . Conse-quently χ / ∈ pX p ( G k ). It follows that χ , ..., χ n are linearly independentmodulo pX p ( G k ). Since n can be chosen arbitrarily large, the proof iscomplete. (cid:3) Since a general cyclic extension
K/k does not introduce new roots ofunity, we have actually shown that a general cyclic extension
K/k with p | [ K : k ] satisfies b p ( K/k ) = s p ( k ). This means that the separation ofcases (I) and (II) is in principle governed only by the number of rootsof unity contained in k . 5. Examples
We determine fibers that contain noncrossed products of degree 8and 9.
Example 5.1. a) Let k = Q . To answer the question “For which qua-dratic number fields Q ( χ ) are there noncrossed products of index 8 inthe fiber over χ ?” we apply Theorem 2.2 with m = 4. By (i) and (ii),these are precisely the quadratic fields Q ( χ ) with ht ( χ ) + s ( χ ) < ( χ ) = 0 and s ( χ ) = 1. This condition is satisfied,for instance, by the third cyclotomic field, Q ( √− Q ( √−
3) is the field of this kind with smallestdiscriminant and smallest conductor. Example 8.4 in Section 8 dis-cusses for which α ∈ Br( Q ) the division algebra in α + χ is a noncrossedproduct of index 8 and computes densities.b) Similarly, the cubic number fields Q ( χ ) that allow noncrossedproducts of index 9 in the fiber over χ are precisely the ones satisfyinght ( χ ) + s ( χ ) <
1, or, equivalently, ht ( χ ) = 0 and s ( χ ) = 0. Thesmallest such field is the cubic subfield of the 7-th cyclotomic field (cf.Example 4.3 with p = 3 and q = 7).We shall also give two small examples in finite characteristic.c) Let k = F ( t ) and let k ( χ ) = F ( t )( √ t ). The prime t is tamelyramified in k ( χ ) and the residue field of k with respect to t has 3elements. So Proposition 4.1 shows ht ( χ ) ≤ − s ( χ ) = 1,there are noncrossed products of index 8 in the fiber over χ .d) Let k = F ( t ) and let k ( χ ) be generated by a root of X + p ( t ) X + p ( t ) where p ( t ) = t + t + 1. (Note that k ( χ ) /k is indeed aGalois extension because the discriminant of a polynomial of the form X + cX + c is always a square in characteristic 2.) Since X + p ( t ) X + p ( t ) is an Eisenstein polynomial for the prime p ( t ), it is irreducibleover k and p ( t ) is tamely ramified in k ( χ ). The residue field of k with respect to p ( t ) has 4 elements, so Proposition 4.1 shows ht ( χ ) ≤ − s ( χ ) = 0 because k ( χ ) /k is cubic. Thus, thereare noncrossed products of index 9 in the fiber over χ . Remark.
Let k be a global field. If p = char k and µ p k thenthere are crossed products that have a noncrossed product p -primarycomponent. Proof.
Let p be as described. Choose a character χ ∈ X ( G k ) of p -power order with s p ( χ ) = 0 and ht p ( χ ) < ∞ . By Theorem 2.2, thereare noncrossed products α + χ of index | χ | p n , where n = ht p ( χ ) + 1.(Note that χ is not exceptional, since p = 2.) Let χ ′ ∈ X ( G k ) such that k ( χ ′ ) = k ( µ p ). Since | χ ′ | is relatively prime to | χ | , we have ht p ( χχ ′ ) =ht p ( χ ) and s p ( χχ ′ ) ≥ s p ( χ ′ ) >
0. Thus, n ≤ ht p ( χχ ′ ) + s p ( χχ ′ ) = b p ( χχ ′ ). The product ( α + χ ) ⊗ (1+ χ ′ ) = α + χχ ′ has index | χ | p n · | χ ′ | = | χχ ′ | p n , so it is a crossed product by Theorem 2.2. (cid:3) Example 5.2.
Let k = Q , p = 3 and let k ( χ ) be the cubic subfieldof the 7-th cyclotomic field (cf. Example 5.1b). Then any noncrossedproduct α + χ of index 9 (of which there are infinitely many) becomesa crossed product when tensored with the quaternion division algebra( − , t Q (( t )) ). 6. Brussel’s Lemma and Galois covers
Let k be a global field. For any α ∈ Br( k ) and χ ∈ X ( G k ) wewrite α + χ for the sum in Br( k ) ⊕ X ( G k ), regarded canonically (de-pending on t ) as a Brauer class over k ( t ) or k (( t )). This means χ isidentified with the class of the cyclic algebra ( k ( χ )( t ) /k ( t ) , σ χ , t ) resp.( k ( χ )(( t )) /k (( t )) , σ χ , t ) defined by χ and α is identified with its restric-tion to k ( t ) resp. k (( t )). Due to Nakayama [13] we have the indexformula ind( α + χ ) = | χ | · ind α k ( χ ) , (6.1)where α k ( χ ) denotes the restriction of α to the cyclic extension k ( χ ).In fact, one commonly proves this formula first over k (( t )) and thenderives it over k ( t ). Brussel’s Lemma.
Let α ∈ Br( k ) and let χ ∈ X ( G k ) . Then α + χ is a crossed product if and only if there is a Galois extension M/k containing k ( χ ) that splits α and has degree [ M : k ( χ )] = ind α k ( χ ) . ONCROSSED PRODUCTS IN WITT’S THEOREM 11
In the case char k = 0 Brussel’s Lemma is [4], Corollary on p.381, butalso holds for arbitrary characteristic (cf. Hanke [7], Theorem 5.20).We start our investigation into the existence of Galois extensions M/k as in Brussel’s Lemma by introducing the following terminology.Let
K/k be a cyclic extension of global fields and let m be a positiveinteger. A Galois m -cover of K/k is an extension field M ⊇ K thatis Galois over k and has degree m over K . Since non-Galois covers arenot considered in this paper, we simply use the term cover and alwaysmean Galois cover. We call a cover cyclic if M/k is cyclic. Using thisterminology, ht p ( K/k ) is the maximal number r such that K/k has acyclic p r -cover, or infinity if no maximal r exists. Moreover, by takingfield composita, for any positive integer m with prime factorization m = Q p p n p , K/k has a cyclic m -cover ⇐⇒ n p ≤ ht p ( K/k ) for all p | m . (6.2)Let P be a prime of K . An m -cover M of K/k is said to have fulllocal degree at P if P is non-archimedian and [ M P : K P ] = m , or if P is real and [ M P : K P ] = (2 , m ), or if P is complex. Let S be a set ofprimes of K . An m -cover M of K/k is said to have full local degree in S if it has full local degree at each prime in S . Throughout the paper, S always denotes a finite set of primes while infinite sets of primes arewritten P .Before establishing the connection between Brussel’s Lemma andGalois covers with full local degree we recall a few essential facts aboutHasse invariants. For an exposition of this material the reader is re-ferred to Pierce [15]. The Brauer group of a global field K is given bythe exact sequence1 −→ Br( K ) inv −→ M P non-archim. Q / Z ⊕ M P real Z / Z P −→ Q / Z −→ , (6.3)where P runs over all primes of K . We write inv P α for the P -component of inv α , which is called the Hasse invariant of α at P .For any α ∈ Br( K ), we call the finite setsupp( α ) := { P | P a prime of K , inv P α = 0 } the support of α . (In the literature supp( α ) is sometimes writtenram( α ) for “ramified primes”.) The index ind α P of the completion α P is equal to the order of inv P α , and ind α is equal to the least commonmultiple of all ind α P . We say α has full local index at P if P is non-archimedian and ind α P = ind α , or if P is real and ind α P = (2 , ind α ),or if P is complex. We define the restricted support of α by (note the underline)supp( α ) := { P ∈ supp( α ) | α has full local index at P } . Unless α = 1, supp( α ) is finite and supp( α ) ⊆ supp( α ). Under exten-sion of scalars to a finite Galois extension M ⊇ K ,inv Q α M = [ M P : K P ] · inv P α (6.4)holds for all primes Q of M dividing P . Lemma 6.1.
Let α ∈ Br( k ) , χ ∈ X ( G k ) and m = ind α k ( χ ) . a) If α + χ is a crossed product then there is an m -cover of k ( χ ) /k with full local degree in supp( α k ( χ ) ) . b) If there is an m -cover of k ( χ ) /k with full local degree in supp( α k ( χ ) ) then α + χ is a crossed product.Proof. If α + χ is a crossed product then, by Brussel’s Lemma, thereis an m -cover M of k ( χ ) /k that splits α k ( χ ) . Let P ∈ supp( α k ( χ ) ) andlet Q be a prime of M with Q | P . By (6.3) and (6.4),0 = inv Q α M = [ M P : k ( χ ) P ] inv P α k ( χ ) . Since α k ( χ ) has full local index at P it follows that M has full localdegree at P . Thus, a) is proved.Conversely, any m -cover of k ( χ ) /k with full local degree in supp( α k ( χ ) )splits α k ( χ ) by (6.3) and (6.4), hence b) also follows from Brussel’sLemma. (cid:3) Quantification over all division algebras of fixed degree in a fiberyields
Proposition 6.2.
Let χ ∈ X ( G k ) . Then all division algebras in thefiber over χ of index | χ | m are crossed products if and only if for allfinite sets S of primes of k ( χ ) there is an m -cover of k ( χ ) /k with fulllocal degree in S .Proof. The “if”-part is immediate from Lemma 6.1 b). The “only if”-part follows from Lemma 6.1 a) and the fact that for any finite S thereexists α ∈ Br( k ) with ind α k ( χ ) = m and supp( α k ( χ ) ) ⊇ S (this is aconsequence of (6.3)). (cid:3) Remark.
By negation of Proposition 6.2, the fiber over χ containsnoncrossed products of index | χ | m if and only if there is a finite set S of primes of k ( χ ) such that no m -cover of k ( χ ) /k has full local degreein S . ONCROSSED PRODUCTS IN WITT’S THEOREM 13
We use Proposition 6.2 to reformulate Theorem 2.2 in terms of Galoiscovers. Let
K/k be a cyclic extension of global fields. For each prime p , define b p := b p ( K/k ) as in Definition 2.1. Let m be a positive integerwith prime factorization m = Q p p n p . Then part (i) of Theorem 2.2 isequivalent to Theorem 6.3. If n p ≤ b p for all prime factors p of [ K : k ] then forany finite set S of primes of K there is an m -cover with full local degreein S . Theorem 6.4 below implies part (ii) of Theorem 2.2. (This is clearexcept for the density statement. It is the purpose of Section 8 belowto show that Theorem 6.4 also implies the density statement.)
Theorem 6.4. If n p > b p for some prime factor p of [ K : k ] thenthere are non-archimedian primes P , P , P in K such that there isno m -cover of K/k with full local degree in { P , P , P } . In fact, thereare infinite sets P , P , P of non-archimedian primes of K such thatfor any ( P , P , P ) ∈ P × P × P there is no m -cover of K/k withfull local degree in { P , P , P } . Proofs
We begin with Theorem 6.3. We can assume m to be a prime powerbecause the compositum of p n p -covers with full local degree in S isclearly an m -cover with full local degree in S . So let m = p n ( p prime).We can further assume that [ K : k ] is a p -power, because any p n -cover of K p /k yields a p n -cover of K/k . Under these assumptions Theorem 6.3reads:
Theorem 7.1.
Let p be a prime and let [ K : k ] be a p -power. For any n ≤ b p and any finite set S of primes of K there is a p n -cover of K/k with full local degree in S . Proposition 7.2.
Let p be a prime and let [ K : k ] be a p -power. Forany n ≤ ht p ( K/k ) + s p ( K ) and any finite set S of primes of K thereis a p n -cover of K/k with full local degree in S .Proof. Write n as n = n ′ + n ′′ with n ′ ≤ ht p ( K/k ) and n ′′ ≤ s p ( K ).Since n ′ ≤ ht p ( K/k ), there is a cyclic p n ′ -cover M ′ of K/k . We canassume M ′ to have full local degree in S (this is a standard argument,see e.g. [8, Proposition 2] for details). Now, we invoke the main theoremof [8]: since M ′ contains all p n ′′ -th roots of unity there is a p n ′′ -cover M of M ′ /k with full local degree in S (more precisely: in the set ofprimes of M ′ dividing a prime in S ). The field M is clearly a p n -coverof K/k with full local degree in S . (cid:3) Proof of Theorem 7.1 if
K/k is non-exceptional or p = 2 . Since we have b p = ht p ( K/k ) + s p ( K ) the assertion of Theorem 7.1 isexactly Proposition 7.2. (cid:3) The proof of Theorem 7.1 if
K/k is exceptional and p = 2 is post-poned to Section 12. So far, we have completed the proof of Theorem6.3 when K/k is non-exceptional or m is odd.We turn to Theorem 6.4, starting with a reduction of the proof toprime powers m . Lemma 7.3.
Let m be a positive integer. If char k = p > then write m = p n m with p ∤ m ; if char k = 0 then set m = m .There are infinitely many primes P of K such that any m -cover of K/k with full local degree at P contains an m ′ -cover of K/k (with fulllocal degree at P ) that is abelian over K with m | m ′ .Proof. By Chebotarev’s density theorem (cf. page 7) there are infinitelymany non-archimedian primes P of K that split completely in K ( µ m ).Since char k ∤ m , we can assume ( N ( P ) , m ) = 1 for all of them,hence N ( P ) ≡ m ) by (3.4) on page 6. Let P be such aprime and let M be an m -cover of K/k with full local degree at P .Then N := Gal( M/K ) ∼ = Gal( M P /K P ). If char k = p then let N bethe maximal normal p -subgroup of N ; if char k = 0 then let N = 1.Let W be the wild inertia group of P in M/K . If char k = 0 then W is trivial; if char k = p then W is normal in N . In any case, W ⊆ N .Let e be the tame ramification index of P in M/K . Then e | m , so N ( P ) ≡ m ) implies N ( P ) ≡ e ). Hence, by (3.3)on page 6, N/W is abelian, so
N/N is abelian.Let M be the fixed field of N . Since N is a characteristic subgroupof N , N is normal in Gal( M/k ). So M is Galois over k and abelianover K . Clearly, m divides m ′ := | N : N | = [ M : K ], so the proof iscompleted. (cid:3) Corollary.
For any positive integer m there are infinitely many primes P of K such that any m -cover M of K/k with full local degree at P contains a p n -cover M ′ of K/k (with full local degree at P ) for eachprimary component p n of m with p = char k .Proof. Let P be a prime as in Lemma 7.3 and suppose M is an m -cover of K/k with full local degree at P . Let M ⊆ M be an m ′ -coverof K/k as in Lemma 7.3. For any prime p , the prime-to- p part N ′ of the abelian group Gal( M /K ) is a characteristic subgroup. SinceGal( M /K ) is normal in Gal( M /k ), also N ′ is normal in Gal( M /k ).This proves that M contains a p n -cover of K/k (with full local degree
ONCROSSED PRODUCTS IN WITT’S THEOREM 15 at P ), where p n is the p -primary component of m ′ . Since the p -primarycomponents of m and m ′ are equal for p = char k , we are done. (cid:3) Let P be the infinite set of primes P in the Corollary. As we willexplain, Theorem 6.4 reduces to Theorem 7.4.
Let p be a prime different from char k . For any positiveinteger n > b p there are infinite sets P , P of primes of K such thatfor any P ∈ P and any P ∈ P there is no p n -cover of K/k with fulllocal degree in { P , P } . We show first that Theorem 6.4 follows from Theorem 7.4. Assume n p > b p for some p dividing [ K : k ]. Then p = char k , for otherwise b p = ∞ . Let n = n p and let P , P be as in Theorem 7.4. Suppose M is an m -cover of K/k with full local degree in { P , P , P } for some( P , P , P ) ∈ P × P × P . By choice of P , the m -cover M containsa p n -cover M ′ of K/k . This contradicts Theorem 7.4 because M ′ hasfull local degree in { P , P } . Thus, M cannot exist and Theorem 6.4is proved.We now turn to the proof of Theorem 7.4. For the rest of this section,the prime p is fixed and different from char k . Define the cyclotomicpart of K/k to be T := K ∩ k ( µ p ∞ ) , where µ p ∞ denotes the group of all p -power roots of unity. Of course, s p ( T ) = s p ( K ). Lemma 7.5.
For any nonnegative integer n there are infinitely manyprimes P of K such that any p n -cover of K/k with full local degree at P is abelian over T = K ∩ k ( µ p ∞ ) .Proof. Since any p n -cover of K/k with full local degree at P is alsoa p n -cover of K/T with full local degree at P , it suffices to assume T = k . We apply Chebotarev’s density theorem to the compositum K · k ( µ p n ). The extension K/k is cyclic and K ∩ k ( µ p n ) = k , so thereare infinitely many non-archimedian primes p of k that are inert in K and split completely in k ( µ p n ). Since p = char k , we can assume p ∤ N ( p ) for all of them, hence N ( p ) ≡ p n ) by (3.4).Suppose M is a p n -cover of K/k with full local degree at a prime P of K dividing such a p . Since P is inert in K/k and M has fulllocal degree at P , we have Gal( M/k ) ∼ = Gal( M p /k p ). Let e be theramification index of p in M/k . Since p is unramified in K/k , we have e | p n . Thus N ( p ) ≡ e ), and (3.3) shows that M p /k p (hence also M/k ) is abelian. (cid:3)
We distinguish two cases. Say
K/k is Case A if k ( µ p sp ( K )+1 ) /k iscyclic and Case B if k ( µ p sp ( K )+1 ) /k is non-cyclic. Proposition 7.6.
Suppose
K/k is Case A. There are infinite sets P , P of primes of K such that for any p n -cover M of K/k with n > s p ( K ) and with full local degree in { P , P } for some P ∈ P and P ∈ P , M contains a cyclic p n − s p ( K ) -cover of K/k .Proof.
Let s = s p ( K ). Let P be an infinite set of primes of K as inLemma 7.5. We apply Chebotarev’s density theorem to the extensions K/k and k ( µ p s +1 ) /k . Since they are both cyclic there are infinitelymany non-archimedian primes p of k that are inert in K as well as in k ( µ p s +1 ). Let P be an infinite set of primes P of K that divide such a p . Since p = char k , we can assume p ∤ N ( P ) for all of them.Suppose M is a p n -cover of K/k with full local degree in { P , P } for some P ∈ P and P ∈ P . Then there is a unique prime Q of M dividing P . Let I be the inertia field and let p e be the ramificationindex of Q over k . Since P is inert in K/k , the field I is clearly acyclic p n − e -cover of K/k . It remains to show e ≤ s . Let p = P ∩ T . Bychoice of P , M/T is abelian, hence also M Q /T p is abelian. Moreover, M Q /T p is tame because p ∤ N ( P ). Thus, N ( p ) ≡ p e ) by(3.3). On the other hand, by (3.4), N ( p ) p s +1 ) because p is inert in the nontrivial extension T ( µ p s +1 ) /T . This proves e ≤ s . (cid:3) Lemma 7.7 (Case B) . If K/k is Case B then k is a number field, p = 2 and s ( K ) > s ( k ) = 1 . Moreover, a) ˜ k = Q ( η s ) where s := s ( K ) , b) T = k ( i ) .Proof. Suppose
K/k is Case B. Then k is a number field, p = 2 and s ( K ) >
1, for otherwise, k ( µ p s K )+1 ) /k would be cyclic. Let s = s ( K ). Since Gal( k ( µ s +1 ) /k ) ∼ = Gal(˜ k ( µ s +1 ) / ˜ k ), we can conclude fromthe diagram (3.1) on page 5 that ˜ k = Q ( η r ) for some r ≤ s . But k ( µ s ) /k is cyclic, so r = s . This proves s ( k ) = 1 and part a). Since i ∈ ˜ K we have ˜ K = Q ( µ s ). Hence T = k · ˜ K = k ( i ). Thus, b) isproved. (cid:3) Proposition 7.8.
Suppose
K/k is Case B. There are infinite sets P , P of primes of K such that any n -cover of K/k with n > s ( K ) and with full local degree in { P , P } for some P ∈ P and P ∈ P contains a cyclic -cover of K/k .Proof.
Let s = s ( K ). By Lemma 7.7 a), k is a number field, p = 2, s > s ( k ), η s ∈ k and η s +1 K . Any non-archimedian prime p ONCROSSED PRODUCTS IN WITT’S THEOREM 17 of k that is inert in K and does not divide 2 has N ( p ) ≡ − s ).Indeed, by (3.5), η s ∈ k implies N ( p ) ≡ ± s ), and by (3.4), µ s ⊂ K and s > s ( k ) imply N ( p ) s ). The extension k ( η s +1 ) /k is quadratic, hence K ∩ k ( η s +1 ) = k . We apply Chebotarev’sdensity theorem to the compositum K · k ( η s +1 ). On the one hand, thereare infinitely many non-archimedian primes p of k not dividing 2 thatare inert in K as well as in k ( η s +1 ). On the other hand, there areinfinitely many non-archimedian primes p of k not dividing 2 that areinert in K and split completely in k ( η s +1 ). Owing to (3.5), we conclude N ( p ) ≡ − s (mod 2 s +1 ) for all p and N ( p ) ≡ − s +1 )for all p . Hence, N ( p ) N ( p ) l (mod 2 n ) for any n > s and any l ∈ N . (7.1)Let n > s and let P , P be the unique primes of K with P i | p i . Assumethere is a 2 n -cover M of K/k with full local degree in { P , P } thatdoes not contain a cyclic 2-cover of K/k . Then p , p have uniqueprime divisors in M , so the notation Z p i ( M/k ) , I p i ( M/k ) , ϕ p i ( M/k ) isunambiguous and Z p i = k . Moreover, I p i ( M/k ) ⊇ K because p i is inertin K . We conclude I p i ( M/k ) = K , for I p i /Z p i is always cyclic and M was assumed not to contain a cyclic 2-cover of K/k . Thus, ϕ p and ϕ p are both generators of Gal( K/k ). Let ϕ p = ϕ l p with l ∈ N . By(3.2), ϕ p i acts on Gal( F/K ) as the N ( p i )-th power map. This meansthat ϕ p acts at the same time as the N ( p )-th power map and as the N ( p ) l -th power map on a cyclic group of order 2 n , contradicting (7.1).The proof is thus completed by choosing, for each i = 1 ,
2, the set P i to be the set of primes of K dividing one of the infinitely many p i asabove. (cid:3) Proof of Theorem 7.4 if
K/k is non-exceptional or p = 2 . Let
K/k be non-exceptional or p = 2 and let n > b p = ht p ( K/k ) + s p ( K ). Theorem 7.4 claims the existence of infinite sets P , P of primesof K such that for any P ∈ P and any P ∈ P there is no p n -coverof K/k with full local degree in { P , P } . We choose P , P dependingon the case of K/k . Case A : Choose P , P as in Proposition 7.6. Then, if there existsa p n -cover with full local degree in { P , P } for some P ∈ P and P ∈ P , we can conclude ht p ( K/k ) ≥ n − s p ( K ). Since this contradictsthe hypothesis n > ht p ( K/k ) + s p ( K ), there is no p n -cover of K/k withfull local degree in { P , P } for any P ∈ P , P ∈ P . Case B : By Lemma 7.7, k is a number field, p = 2, − K and T = k ( i ). Thus, K/k is non-exceptional by asumption, i.e.ht ( K/k ) = 0 or ht ( K/T ) = ht ( K/k ). Subcase ht ( K/k ) = 0: Choose P , P as in Proposition 7.8. Then, ifthere is a 2 n -cover with full local degree in { P , P } for some P ∈ P and P ∈ P , we can conclude ht ( K/k ) >
0, a contradiction.
Subcase ht ( K/T ) = ht ( K/k ): Since every 2 n -cover of K/k is alsoa 2 n -cover of K/T , it suffices to prove the claim for
K/T . But
K/T isCase A because s ( T ) >
1, hence we are done. (cid:3)
The proof of Theorem 7.4 if
K/k is exceptional and p = 2 is post-poned to Section 12. So far, we have completed the proof of Theorem6.4 when K/k is non-exceptional or m is odd. Altogether at this point,Theorem 2.2 is proved when K/k is non-exceptional or m is odd, exceptfor the density statement.8. Density in the Brauer group
In this section we define the notion of density in the Brauer groupof a global field and then prove that Theorem 6.4 implies the densitystatement of Theorem 2.2, thus completing the proof of Theorem 2.2if
K/k is non-exceptional or m is odd.Let k be a global field. In order to measure the “density” of subsets X ⊆ Br( k ) we consider X S := { α ∈ X | supp( α ) ⊆ S } for growing (finite) sets S of primes of k . Definition 8.1.
Suppose X ⊆ Y ⊆ Br( k ) such that X S and Y S arefinite for any finite S . Define d S ( X | Y ) := | X S || Y S | . For any x > S x denote the set of non-archimedian primes of k with absolute norm ≤ x . We write d x ( X | Y ) for d S x ( X | Y ). If the limitlim x →∞ d x ( X | Y ) exists we define d ( X | Y ) := lim x →∞ d x ( X | Y )and call it the natural density of X in Y .For finite intersections,if d ( X i | Y ) = 1 then d ( \ X i | Y ) = 1 . (8.1) Lemma 8.2.
Let
K/k be a given cyclic extension of degree n . Let P be any given infinite set of non-archimedian primes of K . For a fixed ONCROSSED PRODUCTS IN WITT’S THEOREM 19 integer m > consider the sets Y = { α ∈ Br( k ) | ind α K | m } ,X = { α ∈ Br( k ) | ind α K = m and supp( α K ) ∩ P = ∅} . Then d ( X | Y ) = 1 .Proof. For convenience, denote the infinite set { P ∩ k | P ∈ P } alsoby P . Fix a non-archimedian prime p of k that is inert in K . Wecan assume without loss of generality p P because replacing P by P \{ p } makes X smaller.For any prime p of k write n p = [ K p : k p ]. Let S be an arbitraryfinite set of non-archimedian primes of k . The description of Br( k ) bymeans of Hasse invariants (cf. (6.3) and (6.4) on page 11) allows tonaturally identify Y S with ( y ∈ Y p ∈ S n p m Z / Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X p ∈ S y p = 0 ) . Because S is growing it is natural to assume that it contains our fixedprime p . Let S = S ′ ˙ ∪{ p } . Since n p = n , we have an identification Y S = Y p ∈ S ′ n p m Z / Z . For any vector y = ( y p ) p ∈ S ′ ∈ Y S we have y p = inv p y . If ord y p = n p m for some p ∈ S ′ ∩ P then (6.4) implies ind y K = m , in other words y ∈ X S . Hence, Y S \ X S ⊆ ( y ∈ Y p ∈ S ′ n p m Z / Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ord y p < n p m for all p ∈ S ′ ∩ P ) . Counting the vectors in the set on the right yields | Y S \ X S || Y S | ≤ Y p ∈ S ′ ∩ P (cid:18) − ϕ ( n p m ) n p m (cid:19) ≤ Y p ∈ S ′ ∩ P (cid:18) − ϕ ( nm ) nm (cid:19) = (cid:18) − ϕ ( nm ) nm (cid:19) | S ′ ∩ P | . (Note that if a | b , then ϕ ( a ) a ≥ ϕ ( b ) b .)Given ε >
0, choose r sufficiently large such that (1 − ϕ ( nm ) nm ) r < ε and choose x sufficiently large such that | S ′ x ∩ P | ≥ r . We get d x ( X | Y ) = | X S x || Y S x | ≥ − ε. This proves the lemma. (cid:3)
Theorem 8.3.
Let χ ∈ X ( G k ) and let m be a positive integer withprime factorization m = Q p p n p . Suppose char k ∤ m and n p > b p forsome p dividing | χ | . For the sets B := { α ∈ Br( k ) | ind( α + χ ) = | χ | m } ,B := { α ∈ B | α + χ is a noncrossed product } the natural density of B in B exists and d ( B | B ) = 1 . Theorem 8.3 is precisely the density statement of Theorem 2.2 (ii).
Proof.
The proof depends on Theorem 6.4, which has been proved sofar only for non-exceptional
K/k or odd m . However, the proof willshow that Theorem 8.3 holds whenever Theorem 6.4 holds.Let K = k ( χ ). Let P , P , P be infinite sets of non-archimedianprimes of K as in Theorem 6.4. Define Y = { α ∈ Br( k ) | ind α K | m } ,X i = { α ∈ Br( k ) | ind α K = m and supp( α K ) ∩ P i = ∅} for i = 0 , ,
2. By choice of the P i and by Lemma 6.1 a) we have X ∩ X ∩ X ⊆ B , and B ⊆ Y is obvious from (6.1). Thus, it sufficesto show d ( X ∩ X ∩ X | Y ) = 1. By Lemma 8.2, d ( X i | Y ) = 1 for each i .The proof is complete because of (8.1). (cid:3) This completes the proof of Theorem 2.2 if
K/k is non-exceptionalor m is odd. Example 8.4.
Let k = Q and let K = Q ( χ ) = Q ( √−
3) as in Example5.1a. Then s ( χ ) = 1 and ht ( χ ) = 0 (cf. Example 4.3). Thus, Theorem8.3 applies with m = 2 n for all n ≥
2, and we have B = { α ∈ Br( Q ) | ind( α + χ ) = 2 n +1 } = { α ∈ Br( Q ) | ind α K = 2 n } ,B = { α ∈ B | α + χ is a noncrossed product } . Our goal is to describe explicitly a subset of B with density 1 in B .For this purpose, we simply have to determine the sets P , P fromProposition 7.6. Note that K/ Q is case A. The odd primes that areinert in K are the primes ≡ P consists of theprimes that are inert in K and split completely in Q ( µ n ), so P = { q ∈ P | q ≡ n (mod 3 · n ) } if n is even, P = { q ∈ P | q ≡ n +1 (mod 3 · n ) } if n is odd.The set P consists of the primes that are inert in K as well as in Q ( µ ), so P = { q ∈ P | q ≡ − } . ONCROSSED PRODUCTS IN WITT’S THEOREM 21 If α ∈ B satisfies ind q i α K = 4 at some q ∈ P and some q ∈ P then α + χ is a noncrossed product. This occurs with density 1 in B .The remainder of the paper is primarily devoted to the proof ofTheorem 2.2 for exceptional K/k and even m .9. The Local-Global Principle for the height
Theorem 9.1.
Let
K/k be a cyclic extension of local fields. For anyprime-power p r , ht p ( K/k ) ≥ r ⇐⇒ µ p r ( k ) ⊂ N K/k ( K ∗ ) . (In contrast to Albert’s Theorem only those p r -th roots that areactually contained in k are required to be norms here.) Proof.
This is a consequence of local class field theory and can be foundin Neukirch [14, Satz (5.1), p. 83]. (cid:3)
Corollary 9.2.
Let k be a local field with char k = 2 and i k . Thenany finite cyclic extension K/k has ht ( K/k ) ∈ { , ∞} .Proof. Since µ r ( k ) = {± } , the condition µ r ( k ) ⊂ N K/k ( K ∗ ) is inde-pendent of r for all r ≥
1. Thus, by Theorem 9.1, ht ( K/k ) > ( K/k ) = ∞ . (cid:3) Now, let
K/k be a cyclic extension of global fields. Recall that byLemma 4.2, ht p ( K/k ) ≤ ht p ( K p /k p ) for all primes p in k . Definition 9.3.
The global field k is called special with index s if k is a number field with ˜ k = Q ( η s ) for some s ≥ e k p ) ˜ k for allprimes of k . (cf. Geyer-Jensen [6, Definition 2, p. 709])Recall that e k stands for the intersection k ∩ Q ( µ ∞ ). The condition e k p = ˜ k is in fact only required for even primes, for it always holds forodd primes. Theorem 9.4.
Let
K/k be a cyclic extension of global field and let p be a prime. For all r ∈ N , if p = 2 or k is not special or k is specialwith index ≥ r then: ht p ( K/k ) ≥ r ⇐⇒ ht p ( K p /k p ) ≥ r for all primes p in k . (9.1) If k is special with index s then there is an id`ele class c of k , such thatfor all r > s : ht p ( K/k ) ≥ r ⇐⇒ ht p ( K p /k p ) ≥ r for all primes p in k and c is the norm of an id`ele class of K . (9.2) Proof.
Artin-Tate [3], Chapter X, Theorem 6. (Beware of the misprint:“c p r ” should be “a ” and “Lemma 2” should be “Theorem 2”. Further-more, “in the special case” should be understood as “in the special casewith S = ∅ with r > s ”.) (cid:3) Corollary 9.5. If K/k is exceptional then k is special with index s =ht ( K/k ) .Proof. Let
K/k be exceptional. By Definition 2.1, k is a number field, i ∈ K and ht ( K/k ( i )) > ht ( K/k ) >
0. We prove the statementby contradiction, so assume k is not special or special with index s =ht ( K/k ). If k is not special or special with s > ht ( K/k ) then (9.1)holds for all r ∈ N . If k is special with s < ht ( K/k ) then (9.2) appliedwith r = ht ( K/k ) shows that c is a norm. Since c is independent of r ,(9.1) holds for r := ht ( K/k )+1 in both cases. Hence, there is a prime p in k such that ht ( K p /k p ) < r . Since 0 < ht ( K/k ) ≤ ht ( K p /k p ) < ∞ ,we conclude from Corollary 9.2 that i ∈ k p . Therefore, ht ( K/k ( i )) ≤ ht ( K p /k ( i ) p ) = ht ( K p /k p ) < r . This means ht ( K/k ( i )) ≤ r − ( K/k ) in contradiction to
K/k being exceptional, so the claim isproved. (cid:3)
Remark.
Let k be special with index s . Then any finite cyclic exten-sion K/k with i ∈ K is Case B (for p = 2). In particular, s = s ( K ). Proof. If i ∈ K then s ( K ) + 1 ≥
3. Since ˜ k = Q ( η s ), the extension k ( µ s K )+1 ) /k is non-cyclic, i.e. K/k is Case B. By Lemma 7.7, s = s ( K ). (cid:3) Embedding Problems and Galois covers
Let
K/k be a Galois extension of fields and let G = Gal( K/k ).Suppose an embedding of K into the separable closure of k is fixed,and let ϕ : G k → G be the canonical surjection where G k denotes theabsolute Galois group of k . An embedding problem for K/k is a diagram G k A E G ❄ ϕ ♣♣♣ ✠ ψ ✲ ✲ ✲ π ✲ (10.1)where the bottom row is a group extension. By a solution to theembedding problem we mean either a homomorphism ψ that makesthe diagram commute or the fixed field of the kernel of such ψ . Wespeak of a proper solution if ψ is surjective, or A ⊂ im ψ . Another ONCROSSED PRODUCTS IN WITT’S THEOREM 23 embedding problem G k B Γ G ❄ ϕ ♣♣♣ ✠ ψ ✲ ✲ ✲ σ ✲ (10.2)is said to dominate (10.1) if there is a commuting diagram1 −−−→ B −−−→ Γ σ −−−→ G −−−→ y y (cid:13)(cid:13)(cid:13) −−−→ A −−−→ E π −−−→ G −−−→ k be a global field. For each prime p in k , write k p for thecompletion and G p for the absolute Galois group of k p . We regard G p asa subgroup of G k . Let K/k be a finite Galois extension, G = Gal( K/k ),and let ϕ : G k → G be the canonical surjection. We set G p := ϕ ( G p )and define ϕ p : G p → G p as a restriction of ϕ . In this way, we haveassociated to (10.1) for each prime p in k a corresponding local diagram G p A E p G p ❄ ϕ p ♣♣♣♣ ✠ ψ p ✲ ✲ ✲ π ✲ (10.3)where E p = π − ( G p ) ⊆ E . For each prime P in K dividing p we canidentify Gal( K P /k p ) with G p . (In fact, G p is one of the decompositiongroups for p in Gal( K/k ).) Thus, (10.3) is regarded as an embeddingproblem for K P /k p . Any global solution ψ gives local solutions ψ p atall p by restricting ψ to G p , but they are not necessarily proper evenif ψ is proper.An m -cover of K/k is by definition the proper solution to an em-bedding problem of the form (10.1) with | A | = m and arbitrary E .For a non-archimedian prime P in K , the m -cover given by ψ has fulllocal degree at P if and only if the corresponding local solution ψ P isalso proper, i.e. if and only if A ⊂ im( ψ P ). For a real prime P in K ,the m -cover given by ψ has full local degree at P if and only if thecorresponding local solution ψ P satisfies | im( ψ P ) | = (2 , m ).A related well-studied topic is embedding problems with local pre-scription , which exactly prescribe local solutions ψ p at finitely manyprimes rather than just local degrees. Even though this problem is more restrictive than ours, we are required to make use of this the-ory in order to deal with exceptional case. The crucial result for thispurpose is Theorem 10.1 below.We regard A as a G k -module through ϕ . Let A ′ = Hom( A, µ ) denotethe G k -module dual to A , where µ is the natural G k -module of all rootsof unity over k . We write k ( A ′ ) for the fixed field of all σ ∈ G k thatfix A ′ pointwise, and we set G ′ := Gal( k ( A ′ ) /k ). Note that k ( A ′ ) ⊆ K ( µ l ) , where l = exp A . (10.4) Theorem 10.1.
Let A be abelian. Suppose that for all subgroups U ≤ G ′ the map H ( U, A ′ ) res → Y σ ∈ U H ( h σ i , A ′ ) (10.5) is injective. If the embedding problem (10.1) has local solutions every-where then it has a global solution with any local prescription at finitelymany primes.Proof. We first recall two facts about embedding problems with abeliankernel from Neukirch [14]. If the map H ( G k , A ) res → Y p H ( G p , A ) (10.6)is injective then we have a local-global principle saying: there is a globalsolution if and only if there are local solutions everywhere [14, (2.2)].If there is a global solution and the map H ( G k , A ) res → Y p ∈ S H ( G p , A ) (10.7)is surjective then any local prescription at S can be realized [14, (2.5)].Next, we pass to the dual module. For each prime p in k choose in G ′ a decomposition group G ′ p . The map (10.6) is injective if and onlyif H ( G ′ , A ′ ) res → Y p H ( G ′ p , A ′ ) (10.8)is injective [14, (4.5)]. Furthermore, the map (10.7) is surjective if H ( G ′ p , A ′ ) res → Y σ ∈ G ′ p H ( h σ i , A ′ ) (10.9)is injective for all p ∈ S [14, (6.4a)].Now, (10.9) is injective for each p by hypothesis. Due to Cheb-otarev’s density theorem, every σ ∈ G ′ lies in G ′ p for some p . So theinjectivity of (10.8) follows from the hypothesis for U = G ′ . Thiscompletes the proof. (cid:3) ONCROSSED PRODUCTS IN WITT’S THEOREM 25
Theorem 10.1 holds trivially if G ′ is cyclic. For the easiest non-trivialcase we provide Lemma 10.2.
For any bicyclic group G = h σ i × h τ i and G -module M (multiplicatively written), the kernel of the map H ( G, M ) −→ H ( h σ i , M ) × H ( h τ i , M ) × H ( h στ i , M ) is isomorphic to the group Q defined as follows: For any H ≤ G let M H denote the H -invariants. Consider the norm map N σ : M h τ i −→ M G .Then Q := (ker( N σ ) ∩ M σ − ∩ M στ − ) / ( M h τ i ) σ − . Proof.
Let l : G −→ M be a 1-cocycle. We assume that l is normalized( l = 1). Of course, l is fully determined by l σ , l τ and the cocycleconditions. If l is in the kernel then we can further assume w.l.o.g.that l τ = 1. Let x = l σ ∈ M , the only remaining parameter. Thecocycle conditions imply x = l στ = l τσ = x τ , so x ∈ M h τ i , as wellas N σ ( x ) = 1. Since the restrictions of l to h σ i and h στ i also split,we have x ∈ M σ − and x ∈ M στ − . Conversely, any such x actuallydefines a 1-cocycle (with l τ = 1) because N σ ( x ) = 1.Finally, l is split if and only if there exists a ∈ M with a τ − = l τ = 1,i.e. a ∈ M h τ i , and a σ − = l σ = x . This is the assertion. (cid:3) We point out that in the setup of the following section,
K/k is alwayscyclic. By (10.4), k ( A ′ ) ⊆ K ( µ l ), l = exp A . Hence, G ′ is cyclic orbicyclic unless char k = 0 , | l, ˜ k is real and i K . As we will notencounter this case, Lemma 10.2 is sufficient for our purposes.11. Metacyclic -groups A notable exception in the classification of metacyclic p -groups isthe fact that the following two presentations give isomorphic groups(cf. [12, Theorem 22]): h a, c | a s +1 = 1 , c t = a s , a c = a − i∼ = h a, c | a s +1 = 1 , c t = a s , a c = a − s i , for all s, t ≥
2. Indeed, an isomorphism from left to right is establishedby mapping a to ac t − .Let 1 ≤ l < t and let C l be a cyclic group of order 2 l with generator ρ . Then the isomorphism above is even an isomorphism between group extensions1 −→ h a, c l i −→ h a, c | a s +1 = 1 , c t = a s , a c = a − i −→ C l −→ y y (cid:13)(cid:13)(cid:13) −→ h a, c l i −→ h a, c | a s +1 = 1 , c t = a s , a c = a − s i −→ C l −→ , where c maps to ρ in both rows. We denote this extension by1 → A l → E s,t → C l → . (11.1)and choose for E s,t in the sequel the presentation E s,t = h a, c | a s +1 = 1 , c t = a s , a c = a − i . Clearly, | E s,t | = 2 s + t +1 . The center, commutator subgroup and socleare h c i , h a i and h a s i respectively. For each 1 ≤ l < t , the kernel A l is abelian and decomposes as follows into cyclic factors. Case t − l ≤ s : A l = h a i × h a l − ( t − s ) c l i , exp A l = 2 s +1 . Case t − l ≥ s : A l = h c l i × h ac t − s i , exp A l = 2 t +1 − l .We now turn to investigate the embedding problem defined by (11.1).Let K/k be any cyclic extension of order 2 l ( l ≥
1) with Gal(
K/k ) = h ρ i . We write C l for Gal( K/k ) and consider the embedding problem G k A l E s,t C l ❄ ϕ ♣♣♣♣ ✠ ψ ✲ ✲ ✲ ✲ (11.2)We first point out that (11.2) is dominated in two ways by a splitembedding problem. Indeed, defining for all s, t ≥ s,t = h a, c | a s +1 = 1 , c t = 1 , a c = a − i , ∆ s,t = h a, c | a s +1 = 1 , c t = 1 , a c = a − s i , one obtains a dominating embedding problem for (11.2) by replacingthe group extension by either1 → h a, c l i → Γ s,t +1 → C l → , (11.3)or 1 → h a, c l i → ∆ s,t +1 → C l → . (11.4)(Here, c maps to ρ in all three group extensions.) Lemma 11.1.
Let k be any field with char k = 0 and i k . Let Gal(
K/k ) = C l with l ≥ and let i ∈ K . If k has more than one qua-dratic extension then (11.2) has a proper solution for all s < s ( k ( i )) and all t < ht ( K/k ) + l . ONCROSSED PRODUCTS IN WITT’S THEOREM 27
Proof.
Note that Γ s,t +1 , ∆ s,t +1 are semi-direct products h a i ⋊ h c i re-spectively with ord a = 2 s +1 and ord c = 2 t +1 . Let h = t + 1 − l . Since h ≤ ht ( K/k ) there is a cyclic 2 h -cover L of K/k . We have [ L : k ] =2 t +1 . Choose any a ∈ k with √ a k ( i ). Then M := L ( s +1 √ a ) is asolution field for either (11.3) or (11.4). Indeed, since µ s +1 ∈ k ( i ) ⊆ L ,the extension M/k is Galois and
M/L is a Kummer extension. Theaction of Gal(
L/k ) = h c i on Gal( M/L ) = h a i is given by the actionof Gal( L/k ) on µ s +1 . Since µ s +1 ⊂ k ( i ), this action is of order 2.The possible actions of this order are ζ ζ q for q = − , − s or1 + 2 s . Since i is not fixed, q is either − − s . We have thusshown that M is a proper solution for one of the dominating embeddingproblems. (cid:3) Lemma 11.2.
Let k be a number field with ˜ k = Q ( η s ) , s ≥ . Let Gal(
K/k ) = C l with l ≥ and let i ∈ K . For any t > l , Theorem 10.1applies to the embedding problem (11.2) .Proof. We give the proof only for t − l ≤ s . The case t − l ≥ s is similar.Set h := t − l >
0. (The example will be used only once, in the proof ofProposition 12.1, and this use is for t − s = l , i.e. h = s .) For t ≤ s + l we have E s,t = h a, c | a s +1 = 1 , c l + h = a s , a c = a − i ,A l = h a, c l i = h a i × h b i with b := a s − h c l , exp A l = ord a = 2 s +1 , ord b = 2 h . Note b c = a − s − h +1 b . The action ofGal( K/k ( i )) = h ρ i on A is trivial because c lies in the center of E s,t .Thus, k ( A ′ ) ⊆ k ( µ s +1 ).We continue to determine k ( A ′ ) and G ′ exactly. Let ζ = ζ s +1 . Then A ′ = h a ∗ i × h b ∗ i where a ∗ : a ζ , b ,b ∗ : a , b ζ s +1 − h . The action of G k on A ′ is through G := Gal( k ( µ s +1 ) /k ). This groupis the Klein 4-group, generated for instance by σ, τ with σ : ζ ζ − and τ : ζ ζ s +1 . Since σ restricts to a generator of Gal( k ( i ) /k ), itacts on A like ρ does. Hence: σ ( a ∗ )( a ) = σ ( a ∗ ( a c )) = σ ( a ∗ ( a − )) = σ ( ζ − ) = ζ ,σ ( a ∗ )( b ) = σ ( a ∗ ( b c )) = σ ( a ∗ ( a − s +1 − h b )) = σ ( ζ − s +1 − h ) = ζ s +1 − h ,σ ( b ∗ )( a ) = σ ( b ∗ ( a c )) = σ ( b ∗ ( a − )) = 1 ,σ ( b ∗ )( b ) = σ ( b ∗ ( b c )) = σ ( b ∗ ( a − s +1 − h b )) = σ ( ζ s +1 − h ) = ζ − s +1 − h , i.e. σ ( a ∗ ) = a ∗ b ∗ , σ ( b ∗ ) = b ∗− . Since τ restricts to the identity on k ( i ),it acts trivially on A . Hence: τ ( a ∗ )( a ) = τ ( a ∗ ( a )) = τ ( ζ ) = ζ s +1 ,τ ( a ∗ )( b ) = τ ( a ∗ ( b )) = 1 ,τ ( b ∗ )( a ) = τ ( b ∗ ( a )) = 1 ,τ ( b ∗ )( b ) = τ ( b ∗ ( b )) = τ ( ζ s +1 − h ) = ζ s +1 − h , i.e. τ ( a ∗ ) = ( a ∗ ) s +1 , τ ( b ∗ ) = b ∗ . Furthermore: στ ( a ∗ ) = ( a ∗ ) s +1 b ∗ , στ ( b ∗ ) = b ∗− . Neither σ nor τ acttrivially on A , hence k ( A ′ ) = k ( µ s +1 ) and G ′ = G = h σ, τ i .It suffices to check the injectivity of (10.5) for U = G ′ , the onlynon-cyclic subgroup of G ′ . Easy calculations yield A ′ σ − = h b ∗ i and A ′ στ − = h a ∗ s b ∗ i , hence A ′ σ − ∩ A ′ στ − = h b ∗ i . On the otherhand, b ∗ ∈ A ′ h τ i and ( b ∗ ) σ − = b ∗− . This shows that Q vanishesin Lemma 10.2 with A ′ for M and G ′ for G . (cid:3) Proofs (exceptional case)
This section completes the proof of Theorem 2.2 for exceptional
K/k and even m . It remains to prove Theorems 7.1 and 7.4 for K/k exceptional and p = 2. So let K/k be a cyclic extension ofglobal fields that is exceptional, i.e. k is a number field, i ∈ K andht ( K/k ( i )) > ht ( K/k ) >
0. We begin with Theorem 7.1.
Proposition 12.1.
Supppose
K/k is exceptional and [ K : k ] is a -power. Let n = ht ( K/k ) + s ( K ) + 1 . For any finite set S of primesof K there is a n -cover with full local degree in S .Proof. Let Gal(
K/k ) = C l = h ρ i . By Corollary 9.5 on page 22 and theRemark following it, k is special with index s = s ( K ). By Lemma 11.2,Theorem 10.1 applies to the embedding problem (11.2) for all t > l .Setting t := l + ht ( K/k ), any solution to (11.2) is a 2 n -cover of K/k because | A l | = 2 s + t +1 − l = 2 s +ht ( K/k )+1 = 2 n . Using Theorem 10.1,it remains to prove the existence of proper local solutions at all non-archimedian primes p in k . (All archimedian primes become complexin K since i ∈ K .) So let p be a non-archimedian prime in k and let G p = h ρ g i , 0 ≤ g ≤ l . Case 1: g = 0. In this case, G p = C l and E p = E s,t , so thelocal embedding problem is identical to (11.2) with K/k replaced by K p /k p . According to Lemma 11.1, it remains to verify s < s ( k p ( i ))and t < ht ( K p /k p ) + l .Since g = 0, p is not split in k ( i ), i.e. k p ( i ) /k p is quadratic. Onthe other hand, k p ( µ s +1 ) /k p is at most quadratic, for k is special with ONCROSSED PRODUCTS IN WITT’S THEOREM 29 index s . Thus, µ s +1 ⊂ k p ( i ), i.e. s ( k p ( i )) > s . By Lemma 4.2, wehave ht ( K p /k p ) ≥ ht ( K/k ) >
0. Since i k p , Corollary 9.2 impliesht ( K p /k p ) = ∞ . Case 2: g >
0. In this case, E p = h a, c g i is abelian, for c lies inthe center of E s,t . By Corollary 9.5 we have s = ht ( K/k ), i.e. t = l + s .Thus, ord c g = 2 s +1+( l − g ) ≥ s +1 = ord a , so E p = h c g i × h ac l i ∼ = C s +1+( l − g ) × C s . A solution to G p h a, c l i h c g i × h ac l i h ρ g i ❄ ϕ p ♣♣♣♣♣♣♣♣ ✙ ψ p ✲ ✲ ✲ ✲ (12.1)is then obtained by composing a cyclic 2 s +1 -cover of K p /k p with aKummer extension of k p of degree 2 s . (Note that 2 s +1 is the order of c l .) It remains to verify ht ( K p /k p ) ≥ s + 1 and s ( k p ) ≥ s .Since g > p splits in k ( i ), i.e. i ∈ k p . By Lemma 4.2 on page 8 andthe fact that K/k is exceptional we have ht ( K p /k p ) = ht ( K p /k p ( i )) ≥ ht ( K/k ( i )) > ht ( K/k ) = s . Finally, η s , i ∈ k p imply s ( k p ) ≥ s . (cid:3) Proof of Theorem 7.1 for
K/k exceptional and p = 2 . Since we have n ≤ b = ht ( K/k ) + s ( K ) + 1, the assertion of Theo-rem 7.1 follows from Propositions 7.2 and 12.1 together. (cid:3) We now turn to Theorem 7.4.
Proposition 12.2.
Suppose
K/k is exceptional. There are infinitesets P , P of primes of K such that for any n -cover M of K/k with n > s ( K ) and with full local degree in { P , P } for some P ∈ P and P ∈ P , M contains a cyclic n − s ( K ) − -cover of K/k .Proof.
The proof is a modification of the proof of Proposition 7.6. Let s = s ( K ). By Proposition 7.6, we can assume K/k is Case B, sinceany cyclic 2 n − s -cover contains a cyclic 2 n − s − -cover of K/k .Let P be an infinite set of primes of K as in Lemma 7.5. We applyChebotarev’s density theorem to the extensions K/k and k ( η s +1 ) /k .Since they are both cyclic ( k ( η s +1 ) /k is quadratic by Lemma 7.7),there are infinitely many non-archimedian primes p of k that are inertin K as well as in k ( η s +1 ). Let P be an infinite set of primes P of K that divide such a p . Since char k = 2 ( k is a number field), we canassume N ( P ) is odd for all of them.Suppose M is a 2 n -cover of K/k with full local degree in { P , P } for some P ∈ P and P ∈ P . Then there is a unique prime Q of M dividing P . Let I be the inertia field and let 2 e be the ramification index of Q over k . Since P is inert in K/k , the field I is clearly a cyclic2 n − e -cover of K/k . It remains to show e ≤ s + 1. Let p = P ∩ T . Bychoice of P , M/T is abelian, hence also M Q /T p is abelian. Moreover, M Q /T p is tame because N ( P ) is odd. Thus, N ( p ) ≡ e ) by(3.3) on page 6.On the other hand, since η s ∈ k (Lemma 7.7) we have N ( p ) ≡ ± s ) by (3.5). Furthermore, since p is inert in k ( η s +1 ), we have N ( p )
6≡ ± s +1 ) again by (3.5). It follows that N ( p ) ≡ ± s (mod 2 s +1 ). Since T /k is quadratic (Lemma 7.7) and p is inert in T , N ( p ) = N ( p ) ≡ s +1 s +2 ). This together with N ( p ) ≡ e ) shows e ≤ s + 1. (cid:3) Proof of Theorem 7.4 for
K/k exceptional and p = 2 . Let n > b ( K/k ) = ht ( K/k ) + s ( K ) + 1. Choose P , P as inProposition 12.2. Then, if there exists a 2 n -cover with full local de-gree in { P , P } for some P ∈ P and P ∈ P , we can concludeht ( K/k ) ≥ n − s ( K ) −
1. Since this contradicts the hypothesis n > ht ( K/k ) + s ( K ) + 1, there is no 2 n -cover of K/k with full localdegree in { P , P } for any P ∈ P , P ∈ P . (cid:3) This completes the proof of Theorem 2.2 also for exceptional
K/k and even m . References [1] A. Albert.
Modern Higher Algebra . Univ. of Chicago Press, Chicago, 1937.[2] S. Amitsur. On central division algebras.
Israel J. Math. , 12:408–420, 1972.[3] E. Artin and J. Tate.
Class Field Theory . Benjamin, New York Amsterdam,1967.[4] E. Brussel. Noncrossed products and nonabelian crossed products over Q ( t )and Q (( t )). Amer. J. Math. , 117:377–393, 1995.[5] E. Brussel. Non-crossed products over function fields.
Manuscripta Math. ,107(3):343–353, 2002.[6] W.-D. Geyer and C. Jensen. Embeddability of quadratic extensions in cyclicextensions.
Forum Math. , 19(4):707–725, 2007.[7] T. Hanke.
A Direct Approach to Noncrossed Product Division Algebras . Dis-sertation, Universit¨at Potsdam, 2001.[8] T. Hanke. On absolute Galois splitting fields of central-simple algebras.
J.Numb. Th. , 126:74–86, 2007.[9] T. Hanke. The local-global principle for the height of a cyclic extension andits special case. Preprint, 2009.[10] I. Kaplansky.
Infinite abelian groups . University of Michigan Press, Ann Arbor,1954.[11] S. Lang.
Algebraic number theory , volume 110 of
Graduate Texts in Mathemat-ics . Springer-Verlag, New York, second edition, 1994.[12] S. Liedahl. Presentations of metacyclic p -groups with applications to K -admissibility questions. J. Algebra , 169(3):965–983, 1994.
ONCROSSED PRODUCTS IN WITT’S THEOREM 31 [13] T. Nakayama. Divisionsalgebren ¨uber diskret bewerteten perfekten K¨orpern.
J. Reine Angew. Math. , 178:11–13, 1937.[14] J. Neukirch. ¨Uber das Einbettungsproblem der algebraischen Zahlentheorie.
Inv. math. , 21:59–116, 1973.[15] R. Pierce.
Associative Algebras . Springer-Verlag, New York, 1982.[16] A. Weil.
Basic number theory . Springer-Verlag, New York, third edition, 1974.Die Grundlehren der Mathematischen Wissenschaften, Band 144.[17] E. Witt. Schiefk¨orper ¨uber diskret bewerteten K¨orpern.
J. Reine Angew.Math. , 176:153–156, 1936.
Department of Mathematics, Technion — Israel Institute of Tech-nology, Haifa, 32000, Israel
E-mail address : [email protected] Current address : Lehrstuhl D f¨ur Mathematik, RWTH Aachen, Templergraben64, D-52062 Aachen, Germany
Department of Mathematics, Technion — Israel Institute of Tech-nology, Haifa, 32000, Israel
E-mail address ::