Splitting Behavior of S n -Polynomials
aa r X i v : . [ m a t h . N T ] M a r SPLITTING BEHAVIOR OF S n -POLYNOMIALS JEFFREY C. LAGARIAS AND BENJAMIN L. WEISS
Abstract.
We analyze the probability that, for a fixed finite set ofprimes S , a random, monic, degree n polynomial f ( x ) ∈ Z [ x ] with coef-ficients in a box of side B satisfies: (i) f ( x ) is irreducible over Q , withsplitting field K f / Q over Q having Galois group S n ; (ii) the polynomialdiscriminant Disc ( f ) is relatively prime to all primes in S ; (iii) f ( x ) hasa prescribed splitting type (mod p ) at each prime p in S .The limit probabilities as B → ∞ are described in terms of values ofa one-parameter family of measures on S n , called z -splitting measures,with parameter z evaluated at the primes p in S . We study propertiesof these measures. We deduce that there exist degree n extensions of Q with Galois closure having Galois group S n with a given finite setof primes S having given Artin symbols, with some restrictions on al-lowed Artin symbols for p < n . We compare the distributions of thesemeasures with distributions formulated by Bhargava for splitting prob-abilities for a fixed prime p in such degree n extensions ordered by sizeof discriminant, conditioned to be relatively prime to p . Introduction
By an S n -polynomial we mean a degree n monic polynomial f ( x ) ∈ Z [ x ]whose splitting field K f / Q , obtained by adjoining all roots of f ( x ) has Galoisgroup S n . It is well known that with high probability a “random” degree n monic polynomial with integer coefficients independently drawn from a box[ − B, B ] n is irreducible and is an S n -polynomial. In 1936 van der Waerden[44] showed that this probability approaches 1 as the box size B → ∞ .For such a polynomial, adjoining one root of f ( x ) gives an S n -number field.Later authors obtained quantitative versions giving explicit bounds for thecardinality of the exceptional set; see Section 5.1.This paper considers a refinement of this problem: to study the set ofpolynomials with coefficients in a box [ − B, B ] n which are S n -polynomialsprescribed to have a given splitting behavior at a given finite set of primes { p k : 1 ≤ k ≤ r } . It shows the existence of limiting splitting densitiesas B → ∞ , conditional on the discriminant Disc( f ) of the polynomial Date : March 13, 2015.1991
Mathematics Subject Classification.
Primary 11R09; Secondary 11R32, 12E20,12E25.The first author was partially supported by NSF grants DMS-1101373 and DMS-1401224. f being relatively prime to Q ri =1 p i . This conditioning imposes a non-ramification condition, requiring the polynomials to have square-free fac-torizations (mod p i ) with 1 ≤ i ≤ r . This conditioning has two importantconsequences:(1) The square-free assumption permits the limiting splitting densities tobe interpreted as a set of probability distributions on the symmetricgroup S n , which depend on the prime p . These distributions areconstant on conjugacy classes of S n .(2) The resulting limit of distributions possess an interpolation prop-erty as p varies. The splitting densities are the values at z = p of a one-parameter family of complex-valued measures ν ∗ n,z on thesymmetric group S n which we call z -splitting measures. The inter-polation property is: the values ν ∗ n,z ( g ) on fixed elements g ∈ S n arerational functions in the parameter z .These limiting splitting densities at z = p have a simple origin. They areinherited from corresponding densities for splitting of polynomials in p -adicfields recently studied by the second author [45], which in turn arise fromsplitting probabilities for polynomials over finite fields. The latter prob-abilities are evaluated by counting the monic polynomials over F p havingvarious square-free factorization types in F p [ X ], for which there are explicitcombinatorial formulas.The first contribution of this paper is to introduce and study the z -splitting measures on S n , and show that for parameter values z = p theyare limiting splitting distributions for S n -polynomials above as the box size B → ∞ . A second contribution is to to compare and contrast the limitingprobabilities of the model of this paper to a recent probability model ofBhargava [2], which considers algebraic number fields of degree n , called S n -number fields, whose normal closure has Galois group S n . Bhargava’s modelconcerns limiting splitting probabilities of a fixed prime p taken over S n -number fields having discriminant bounded by a parameter D , as D → ∞ .The interesting feature is that the limiting probabilities of the two modelsdo not agree. We now describe these two contributions in more detail.1.1. Existence and properties of z -splitting measures. The paper di-rectly defines the z -splitting measures as rational functions of z by a com-binatorial formula given in Definition 2.2, and studies their basic propertiesin Section 4. Only later in the paper do we show that for z a prime powerthese p k -splitting densities coincide with the limiting densities for splittingof S n -polynomials, doing this for k = 1 in Section 5 over the rational field Q and for general k in Section 6 for polynomials with coefficients over generalnumber fields.The splitting types of a square-free monic polynomial (mod p ) of degree n are described by partitions µ of n , which are identified with conjugacyclasses on the symmetric group S n . For each n ≥ , and for each prime p inSection 2.1 we define p -splitting measures ν ∗ n,p ( · ) on S n which are constant PLITTING BEHAVIOR OF S n -POLYNOMIALS 3 on conjugacy classes C µ of S n . We show the following results, whose precisestatements are given in Section 2.(i) For a fixed prime p , the limiting probabilities as B → ∞ for degree n monic polynomials f ( x ) ∈ Z [ x ] conditioned on p ∤ Disc( f ) to havea given splitting type µ exist and are given by the values ν ∗ n,p ( C µ )(See Theorem 2.4). For a fixed splitting type µ the values ν ∗ n,p ( C µ )as functions of the prime p are interpolated by a rational function R µ ( z ) ∈ C ( z ), where we have R µ ( p ) = ν ∗ n,p ( C µ ) holding for each p . This rational function interpolation property yields a parametricfamily of (complex-valued) measures ν ∗ n,z on S n for z ∈ P ( C ) r { , } ,termed z -splitting measures .(ii) The z -splitting measure is a positive probability measure whenever n is an integer greater than 1 and z = t is a real number greater than n −
1. The uniform distribution on S n is the z -splitting measure for z = ∞ ∈ P ( C ) (See Theorem 2.3).(iii) There exist infinitely many S n -number fields having prescribed split-ting types ( p i , µ i ) at a given finite set of primes S = { p , ..., p r } ,provided that all the splitting types have ν ∗ n,p i ( C µi ) >
0. The lat-ter conditions are satisfied if and only if there exists an S n -numberfield K with a subring of algebraic integers that is a monogenic or-der with discriminant relatively prime to Q i p i . The existence ofone such number field K certifies that the associated probability ν ∗ n,p i ( µ i ) > n ≥ exceptional pairs ( p i , µ i ) hav-ing ν ∗ n,p i ( C µi ) = 0. The exceptional primes p i necessarily satisfy2 ≤ p i ≤ n −
1, and this set is nonempty for n ≥
3. The exceptionalpairs correspond to the condition that all S n -number fields havingsuch a splitting type ( p i , µ i ) have the prime p i as an essential dis-criminant divisor (a notion defined in Section 2.3) (Theorem 2.6).The phenomenon of essential discriminant divisors was first noted in1878 by Dedekind [14].The z -splitting measures ν ∗ n,z seem of intrinsic interest, and arise in con-texts not considered in this paper. First, the measure ν ∗ n,k may also have aninteresting representation theoretic interpretation for integer values z = k ,viewing the measure as specifying a rational character of S n . The first au-thor will show that this is the case for z = 1, where the measure is a signedmeasure supported on the Springer regular elements of S n [28]. Second,for z = p k these measures arise in a fundamental example in the theory ofrepresentation stability being developed by Church, Ellenberg and Farb [5].[6], see Section 7.2.1.2. Bhargava S n -number field splitting model. The probability modelfor polynomial factorization (mod p ) studied in this paper has strong par-allels with a probability model developed by Bhargava [2] for the splittingof primes in certain number fields K/ Q of degree n . JEFFREY C. LAGARIAS AND BENJAMIN L. WEISS
Bhargava defines an S n -number field K/ Q to be a number field with[ K : Q ] = n whose Galois closure L over Q has Galois group S n . Thus[ L : Q ] = n ! while [ K : Q ] = n . An S n -number field K is a non-Galoisextension of Q for n ≥
3. Bhargava’s probability model takes as its samplespace, with parameter D , the set of all S n -number fields K of discriminant | D K | ≤ D with the uniform distribution; his results and conjectures concernlimiting behavior of the splitting densities at a fixed prime p as D → ∞ ,conditioned on the restriction that the field K be unramified at ( p ), i.e. p ∤ D K , the (absolute) field discriminant of K . He formulates conjecturesabout these limiting distributions for splitting of a fixed prime ( p ) and provesthem for n ≤
5. These conjectures are unproved for n ≥ S n -number fields with S n -polynomials,which relates the two models. Any primitive element θ of an S n -field that isan algebraic integer has θ being a root of an S n -polynomial. Conversely, ad-joining a single root of an S n -polynomial f ( x ) always yields a field K = Q ( θ )that is an S n -extension in Bhargava’s sense. In the case that p ∤ Disc( f ),where Disc( f ) is the polynomial discriminant, the splitting type of the poly-nomial f ( x ) (mod p ) determines the splitting type of the prime ideal ( p ) in K/ Q , and also the Artin symbol [ K f / Q ( p ) ] (which is a conjugacy class in S n ).The probability model of this paper can then be interpreted as studying pairs( K, α ) in which K is an S n -number field, given with an element α ∈ O K such that K = Q ( α ), with a finite sample space specified by size restric-tions on the coefficients that the (monic) minimal polynomial of α satisfies.Bhargava’s model samples fields K with one distribution, while the modelof this paper samples ( K, α ) with another distribution.We discuss Bhargava’s model in detail in Section 3. Our main observationis that the limiting probability distributions of the two models do not agree:the p -splitting measures depend on p , while Bhargava’s measures are theuniform measure on S n , which is independent of p . We also observe thatBhargava’s limit measure, the uniform measure on S n , arises as the p →∞ limit of the z -splitting measures. In Section 3.2 we present a detailedcomparison of the structural features of the models, and identify differences.However we do not have a satisfying conceptual explanation that accountsfor the differences of the limiting probabilities in the two models, and leavefinding one as an open question..1.3. Plan of Paper.
Section 2 states the main results. Section 3 discussesBhargava’s number field splitting model and compares its predicted prob-ability distributions with the model of this paper. Section 4 derives basicproperties of the splitting probabilities. Section 5 obtains the limiting distri-butions of splitting probabilities for polynomials with integer coefficients ina box. These splitting probabilities are essentially inherited from the analo-gous splitting probabilities for random monic polynomials over finite fields,see Section 4.4. We also establish result (iii) above on existence of infinitelymany S n -number fields having given splitting types at a finite set of primes, PLITTING BEHAVIOR OF S n -POLYNOMIALS 5 avoiding exceptional pairs. Section 6 extends the splitting results of thispaper to monic polynomials with coefficients in rings of integers of a fixednumber field, choosing boxes based on a fixed Z -basis of the ring of integers.The answer involves the splitting measures ν ∗ n,q ( C µ ) for q = q = p f , f ≥ z -splitting densities. Notation.
Our notation for partitions differs from Macdonald [33]. Wedenote partitions of n by µ = ( µ , ..., µ k ), with µ ≥ µ ≥ · · · ≥ µ k , whereMacdonald uses λ ; and the multiplicity of part i of µ is denoted c i ( µ ) := |{ j : µ j = i }| , where Macdonald uses m i ( λ ). We sometimes write a partitionof n in bracket notation as µ = h c , c , · · · , n c n i , with only c i = c i ( µ ) > Results
Splitting Measures.
The results in this paper are expressible in termsof a discrete family of probability distributions on the symmetric group S n indexed by q = p k . These distributions belong to a one-parameter family ofcomplex-valued measures on S n , depending on a parameter z ∈ C r { , } given below, which we call z -splitting measures. Restricting the parameterto real values z = t ∈ R r { , } we obtain signed measures of total mass1, and all the parameter values t = q = p k which are prime powers givenonnegative probability measures on S n ; these measures originally arose instatistics involving the factorization of random square-free polynomials over F q [ X ], see Section 4.4. Definition 2.1.
For each degree m ≥ m -th necklace polynomial M m ( X ) by M m ( X ) := 1 m X d | m µ ( d ) X m/d . where µ ( d ) is the M¨obius function.The necklace polynomial takes integer values at integers n , its values atpositive integers have an enumerative interpretation that justifies its name,given in Section 4.1. These polynomials arise in our context because for X = q = p f a prime power, M m ( q ) counts the number of irreducible monicdegree m polynomials in F q [ X ], where F q is the finite field with q elements,see Lemma 4.1.For a given element g ∈ S n , denote its cycle structure (lengths of cycles)by µ = µ ( g ) =: ( µ , µ , . . . µ k ) with µ ≥ µ ≥ ... ≥ µ k . Here we regard µ as an unordered partition of n , though for convenience we have listed itselements in decreasing order, and we denote it µ ⊢ n . The conjugacy classeson S n consist of all elements g with a fixed cycle structure and we denotethem C µ . For a partition µ ⊢ n we let c i = c i ( µ ) := |{ j : µ j = i }| JEFFREY C. LAGARIAS AND BENJAMIN L. WEISS count its number of parts of size i , and we sometimes denote it by the bracketnotation µ = h c , c , · · · , n c n i , with only c i > Definition 2.2.
The z -splitting measure ν n,z ( g ) for g ∈ S n is given by ν ∗ n,z ( g ) := 1 n ! · z n − ( z − n Y i =1 i c i c i ! (cid:18) M i ( z ) c i ( µ ) (cid:19) , (2.1)where for a complex number w we interpret (cid:0) wk (cid:1) := ( w ) k k ! = w ( w − ··· ( w − k +1) k ! .For each fixed g ∈ S n the quantity ν ∗ n,z ( g ) is a rational function of z ,and is well-defined away from the polar set, which is contained in z = 0 , g depends only on its cycletype µ = µ ( g ), so is constant on conjugacy classes C µ of S n . Using the wellknown formula | C µ | = n ! n Y i =1 i − c i ( µ ) c i ( µ )! , (2.2)for the size of conjugacy classes [41, Proposition 1.3.2], we obtain ν ∗ n,z ( C µ ) := X g ∈ C µ ν ∗ n,z ( g ) = 1 z n − ( z − n Y i =1 (cid:18) M i ( z ) c i ( µ ) (cid:19) . (2.3)Properties of these measures are studied in Section 4. The measures aredefined by the right side of (2.3) as complex-valued measures for all z onthe Riemann sphere, excluding z = 0. The definition implies that they havetotal mass one, in the sense that X g ∈ S n ν ∗ n,z ( g ) = 1 . In this paper we restrict to real values z = t , in which case ν ∗ n,t ( g ) in generaldefines a signed measure on S n . In Section 4.5 we prove results specifyingpositive real z -values where the z -splitting measure is nonnegative. In par-ticular we show nonnegativity of the measure holds for all positive integers t = m ≥ Theorem 2.3.
Let n ≥ . The z -splitting measures ν ∗ n,z have the followingproperties, for positive real parameters z = t > . (1) For all real t > n − , one has ν ∗ n,t ( g ) > for all g ∈ S n , For these parameter values ν n,t ( · ) is a probability measure with fullsupport on S n . (2) For integers k = 2 , , ..., n − , one has ν ∗ n,k ( g ) ≥ for all g ∈ S n , so that ν ∗ n,k ( · ) is a probability measure on S n . For these parametervalues this measure does not have full support on S n . It is zero onthe conjugacy class of the identity element C h n i . PLITTING BEHAVIOR OF S n -POLYNOMIALS 7 (3) As t → ∞ through positive real values, one has lim t →∞ ν ∗ n,t ( g ) = 1 n ! . In Section 4.6 we prove a complementary result specifying negative real z -values where the z -splitting measure is nonnegative. In particular, non-negativity holds for all negative integers m ≤ − n all pairs ( p, µ ) with p a prime and ν ∗ n,p ( C µ ) = 0 . Prime Splitting Densities of S n -Polynomials. Let f ( x ) ∈ Z [ x ] bea monic polynomial. Consider for a prime p the splitting of such polynomials(mod p ), viewed in F p [ X ].More generally for q = p f , any monic f ( x ) ∈ F q [ x ] factors uniquely as f ( x ) = Q ki =1 g i ( x ) e i , where the e i are positive integers and the g i ( x ) aredistinct, monic, irreducible, and non-constant. We may define the splittingtype of such a polynomial (following Bhargava [2]) to be the formal symbol µ q ( f ) := (deg( g ) e , deg( g ) e , . . . , deg( g k ) e k )where k is the number of distinct irreducible factors of f ( x ). Here we orderthe degrees in decreasing order. We let T n denote the set of all possibleformal symbols for degree n polynomials, which we call splitting symbols .Thus T = { (111) , (21) , (3) , (1 , (1 ) } . Using this definition, given anymonic f ( x ) ∈ Z [ x ] and any prime p , we may assign to it a splitting type µ p ( f ) ∈ T n .This paper mainly restricts to square-free splitting types , which are thosehaving all e i = 1. We define T ∗ n ⊂ T n to denote the set of such splitting types.Thus T ∗ = { (111) , (21) , (3) } . Each element µ := ( µ , µ , . . . µ k ) ∈ T ∗ n with µ ≥ µ ≥ ... ≥ µ k specifies a partition of n , to which there is associated aunique conjugacy class C µ ⊂ S n . The conjugacy class C µ is the set of allelements of S n whose cycle lengths are equal to the (unordered) numbers µ , . . . , µ k . In this case we will refer also to C µ as a splitting type , and if µ p ( f ) ∈ T ∗ n then we will write µ p ( f ) = C µ .Given any positive integer n and a positive number B we let F n ( B ) denotethe collection of all degree n monic polynomials with integer coefficients, f ( x ) = x n + n − X j =0 c j x j ∈ Z [ x ] , having coefficients bounded by − B < c j ≤ B, for 0 ≤ j ≤ n −
1. Then let F n,B,p be the subset of monic polynomials in F n ( B ) having the followingproperties:(i) The polynomial discriminant (Disc( f ) , p ) = 1 . JEFFREY C. LAGARIAS AND BENJAMIN L. WEISS (ii) All coefficients of f ( x ) are contained in [ − B + 1 , B ]. This impliesthat the polynomial discriminant | Disc( f ) | ≤ (4 B ) n ( n − . (iii) f ( x ) is irreducible over Q and the degree n number field K f = Q ( θ f )generated by one root has normal closure with Galois group S n .The allowed splitting types (mod p ) of polynomials in F n ( B ; p ) are con-strained by the requirement (1) on the discriminant to belong to T ∗ n , i.e. tobe square-free (mod p ). For this case we show: Theorem 2.4. (Limiting Splitting Densities)
Let n ≥ be given. Then:(1) For each prime p , there holds lim B →∞ { f ( x ) ∈ F n ( B ; p ) } { f ( x ) ∈ F n ( B ) } = 1 − p . (2.4) (2) For each (square-free) splitting type µ ∈ T ∗ n , there holds lim B →∞ { f ( x ) ∈ F n ( B ; p ) | ( p ) has splitting type C µ } { f ( x ) ∈ F n ( B ; p ) } = ν ∗ n,p ( C µ ) , (2.5) where ν ∗ n,p is the splitting measure for n with parameter t = p , i.e. ν ∗ n,p ( C µ ) = 1 p n − ( p − n Y i =1 (cid:18) M i ( p ) c i ( µ ) (cid:19) . (2.6)This result is proved in Section 5; it is a special case r = 1 of Theorem 5.2which applies more generally to finite sets S = { p , p , ..., p r } of primes. InSection 6 we give a further generalization of the result to algebraic numberfields.2.3. Existence of S n -Number Fields with Prescribed Prime Split-ting. We also show there are infinitely many S n -number fields with pre-scribed prime splitting at any finite set S of primes, of those types allowedby the splitting measures. The splitting measures impose some extra con-straints associated to the existence of monogenic orders in the S n -numberfields having discriminants relatively prime to given elements. Theorem 2.5.
Let n ≥ be given, let S = { p , ..., p r } denote a finiteset of (distinct) primes, and let U = { µ , ..., µ r } be a prescribed set of (notnecessarily distinct) splitting symbols for these primes. Then the followingconditions are equivalent. (1) The positive measure condition ν ∗ n,p i ( C µ i ) > for ≤ i ≤ r holds. (2) There exists an S n -number field K having the following two proper-ties: (P1) The field K contains a monogenic order O = Z [1 , θ, ...θ n − ] whose discriminant is relatively prime to p p · · · p r . PLITTING BEHAVIOR OF S n -POLYNOMIALS 9 (P2) The Galois closure K spl of K/ Q is unramified at all prime idealsabove those in S and the primes in S have prescribed Artinsymbols h K spl / Q ( p i ) i = C µ i , ≤ i ≤ r. (3) There exist infinitely many S n -number fields K having properties(P1) and (P2). The condition (1) automatically holds when all p i ≥ n , because the prob-ability measure ν ∗ p i ,n then has full support on the group S n . However forprimes 2 ≤ p < n there are restrictions on the allowed splitting behavior.This restriction has to do with the non-existence of monogenic maximal or-ders satisfying (P1) for S n -number fields having specific prime factorizationat small prime ideals. The polynomials f ( x ) generating such fields have essential discriminant divisors , as defined in Cohn [12, Defn. 9.55, Lemma10.44c] and Cohen [8, p. 197]. A famous example due to Dedekind [14] (see[12, Exercise 9.4; Lemma 10.44c]) is an S -number field K for which theprime ideal (2) splits completely in K ; all monogenic orders then have aneven index, and correspondingly ν ∗ , ([1 ]) = 0. However it is known thatinfinitely many S -number fields K exist in which the ideal (2) splits com-pletely in the maximal order. This result follows from results of Bhargavafor n = 3 discussed in Section 3.1. Such fields are not covered by Theorem2.5.Theorem 2.5 allows us to characterize the splitting measures for primevalues t = p ≥ Theorem 2.6.
For p a prime, and a splitting type µ ⊢ n , for fixed n ≥ ,the following three conditions are equivalent. (C1) The splitting measure at t = p has ν ∗ n,p ( C µ ) = 0 . (C2) There are no degree n monic polynomials f ( x ) ∈ Z [ x ] with f ( x )(mod p ) having a square-free factorization of splitting type C µ . (C3) All S n -number fields K in which ( p ) is unramified and has splittingtype µ necessarily have p as an essential discriminant divisor. This result is proved in Section 5.4. The condition (C1) is vacuous for n = 1 ,
2. This theorem provides the easy-to-check criterion (C1) for an S n -number field K to have ( p ) as an essential discriminant divisor, via thesplitting type µ of ( p ) in K . Condition (C2) is a statement about all f ( x ) ∈ Z [ x ]; it does not require f ( x ) to be an S n -polynomial or to be irreducibleover Q . Our proof does not show the existence of even a single field satisfying Related concepts include the inessential discriminant divisor I ( K ) of a field K (Tormhein [42]), also called the non-essential discriminant divisor of K (Sliwa [40]). Here I ( K ) = gcd θ ∈ O K i ( θ ) , where i ( θ ) := [ O K : Z [1 , θ, ..., θ n − ]] . The essential discriminantdivisors are exactly the prime divisors of I ( K ). (C3) for any given pair ( p, µ ) satisfying (C1). Conjecture 3.2 of Bhargavabelow would imply that infinitely many such fields exist, and this conjectureis known to be true for n ≤ Q is replaced by an algebraic number field K . These generalizations are stated as Theorems 6.2 and 6.3, respectively.These generalizations are a more complicated to state, and their proofs arestraightforward, using results of S. D. Cohen [11].3. Bhargava number field splitting model
Recall from Section 1.2 that Bhargava defines an S n -number field K/ Q tobe a number field with [ K : Q ] = n whose Galois closure L over Q has Galoisgroup S n . Bhargava’s number field splitting model has sample space itheset of all S n -number fields K of discriminant | D K | ≤ D with the uniformdistribution, and his results and conjectures concern the limiting behaviorof splitting densities at a fixed prime p as D → ∞ , conditioned on theproperty that the field K be unramified at ( p ), i.e. p ∤ D K , the (absolute)field discriminant of K .3.1. Bhargava’s conjectures for prime splitting in S n -number fields. In 2007 Bhargava [2] formulated conjectures about the splitting of primesaveraged over S n -number fields ordered by the size of their field discrimi-nants. Bhargava developed his conjectures based on the following principle[2, p. 10]:The expected (weighted) number of global S n -number fieldsof discriminant D is simply the product of the (weighted)number of local extensions of Q ν that are discriminant-compatiblewith D , where ν ranges over all places of Q , (finite and infi-nite).In this statement a local extension of Q v means a degree n ´etale algebra E over Q v (not necessarily a field) and discriminant-compatible means thatthe valuation of the discriminant of E matches that of D and that, in thearchimedean case, the signs of the discriminants match. We state two of hisconjectures below in order to later compare them with our results.Firstly, given any positive integer n and a positive number B we let G n ( B )denote the collection of S n -number fields K that have discriminants | D K | ≤ B . Secondly, given any positive integer n , prime p and positive number B we let G n ( B ; p ) denote the collection of all degree n number fields K suchthat:(i) The ideal ( p ) is unramified in K ;(ii) The field discriminant | D K | ≤ B ;(iii) The degree n field K over Q has normal closure having Galois group S n . PLITTING BEHAVIOR OF S n -POLYNOMIALS 11 The first conjecture of Bhargava concerns which fraction of S n -numberfields have field discriminant K relatively prime to p [2, Conj. 1.4]. Conjecture 3.1 (Bhargava) . Fix a prime p and a positive integer n . Then lim B →∞ { K ∈ G n ( B ; p ) } { K ∈ G n ( B ) } = 1 − ρ n ( p ) . (3.1) where ρ n ( p ) is the “probability of ramification,” given by ρ n ( p ) := P n − k =1 q ( k, n − k ) p n − − k P n − k =0 q ( k, n − k ) p n − − k , (3.2) in which q ( k, n ) denotes the number of partitions of k into at most n parts. By convention we set q (0 , n ) = 1 for n ≥ . This distribution ρ n ( p )depends on both n and p and is a rational function of p . For fixed n , ρ n ( p ) = p + O ( p ) as p → ∞ .Bhargava proves Conjecture 3.1 for n ≤
5. For these cases, the proba-bilities are ρ ( p ) = 0 and ρ ( p ) = p +1 , ρ ( p ) = p +1 p + p +1 , ρ ( p ) = p +2 p +1 p + p +2 p +1 , and ρ ( p ) = p +2 p +2 p +1 p + p +2 p +2 p +1 , respectively. In another conjecture, Bhargava[2, Conjecture 5.2] further relates these probabilities to the distribution ofsplitting types in T n having repeated factors.Bhargava’s second conjecture about prime splitting in S n -number fieldsis as follows [2, Conj. 1.3]. Conjecture 3.2 (Bhargava) . Fix a prime p , a positive integer n , and µ ∈ T ∗ n . Then lim B →∞ { K ∈ G n ( B, p ) } | p has Artin symbol in C µ } { K ∈ G n ( B ; p ) } = ν n ( C µ ) , (3.3) where ν n ( · ) denotes the Chebotarev density distribution on conjugacy classesof S n , which is ν n ( C µ ) := | C µ || S n | Conjecture 3.2 predicts that the limiting density exists and agrees withthat predicted by the Chebotarev density theorem for conjugacy classes (see[29], [37, Chap. 7, § S n . This limiting distribution depends on n but isindependent of p . It is proved for n ≤
5. The case n = 3 is deducible fromresults of Davenport and Heilbronn [13], see also Cohen et al [9]. Bhargavaproved the result for n = 4 and n = 5 using his earlier results for discriminantdensity in quartic and quintic fields [1, 3].For a general viewpoint on Bhargava’s conjectures, see Venkatesh andEllenberg [43, Section 2.3]. Bhargava’s conjectures on local mass formulas,were reinterpreted in connection with Galois representations in Kedlaya [26]and further cases were considered by Wood [47, 48]. Random polynomial model versus random number field model.
We compare the distributions for prime splitting in S n number fields in therandom polynomial model against those of the random number field modelgiven in Bhargava’s conjectures. These splitting distributions differ.We summarize the comparison in Table 1. A main feature is that foreach n ≥ p → ∞ limit approaches the uniform density distribution conjectured in Bharagava’smodel.Probability model Random S n -PolynomialModel Random S n -NumberField Model (Bhargava)Sample space Degree n , monic polyno-mials with integer coeffi-cients | c i | ≤ B , generat-ing an S n -number field S n -number fields K withfield discriminant | D K | bounded by D Limit procedure Box size B → ∞ Discriminant D → ∞ Ramification prob-ability at ( p ) Prob[ p divides Disc ( f )]equals p , which is inde-pendent of n Prob[ p divides Disc ( K )]is a quantity θ n ( p )which depends on both n and p (Conjecture3.1)Limiting distri-bution on S n ofsplitting types p -splitting distribution ν ∗ n,p ( C µ ) on conjugacyclasses, whose probabil-ities depend on both n and p Chebotarev distribution ν n ( C µ ) = | C µ | n ! , which isindependent of p (Con-jecture 3.2 )Limit p → ∞ oframification proba-bility 0 0Limit p → ∞ of dis-tribution densities Uniform distribution ν ∗ n, ∞ = ν n on elements of S n Uniform distribution ν ∗ n, ∞ = ν n on elementsof S n Table 1.
Comparison of polynomial splitting model andrandom S n -number field model probabilities. (Conjectures3.1 and 3.2 are theorems for n ≤ S n -number fields K having discriminant prime to p , which depend on the parameter B (resp. D ), and consider the limiting distribution as the corresponding parametergrows. In each model the splitting density is a conditional probability basedon conditioning against an “unramifiedness” condition. There is a differenceof scale in the cutoffs in the B and D parameters between the two models, PLITTING BEHAVIOR OF S n -POLYNOMIALS 13 in that polynomial discriminants D f grow proportionally to B n . Howeverthe limit as the parameters go to infinity, this scale differences play no role.The main differences in the predicted probabilities in the models are thefollowing.(1) In Bhargava’s conjectures the probability of ramification ρ n ( p ) de-pends on both the prime p and the degree n . One has θ n ( p ) := 1 − ρ n ( p ) = 11 + P n − k =1 q ( k, n − k ) p − k , This formula implies that for fixed p and variable n the function ρ n ( p ) increases to the limit ρ ∞ ( p ) := 1 − P (1 /p ) where P ( x ) := ∞ X n =0 p n x n = ∞ Y n =1 ( 11 − x n ) . In contrast, in the random polynomial model the probability of ram-ification p is independent of n , according to Theorem 2.4 (1). Theformula above implies that for fixed n one has ρ n ( p ) = p + O ( p ) as p → ∞ , so both ramification probabilities go to 0 as p → ∞ at thesame rate.(2) In Bhargava’s conjectures the splitting probabilities are independentof both p and n . In contrast, in the random polynomial model theprobabilities ν ∗ n,p ( C µ ) depend on both n and p .What features of the models account for the differing answers in the twomodels? The models themselves have structural differences.(D1) The (irreducible) polynomial f is associated algebraically not withthe ring of integers O F of the field K = Q ( θ ) generated by a root θ of f , but with the particular monogenic order O f = Z [1 , θ, θ , · · · , θ n − ].In particular discriminant Disc( f ) = D K c , where c = [ O K : O f ] isthe index of O f inside f . In particular Disc( f ) may be divisible byprimes which do not divide D K , so the “unramified” conditions ofthe two probability models differ. For some S n -number fields K thering of integers O K is not monogenic. The number of monogenicorders of a given index in the maximal order O K (isomorphism upto an additive shift of a variable) is known to depend on the indexwithin a given field K , cf. Evertse [16].(D2) Many different polynomials in F n ( B ; p ) generate the same S n -numberfield K = K f . Thus each field K that occurs is weighted by thenumber of polynomials in the box that generate it (and which sat-isfy the discriminant co-primeness condition). The weights dependin a complicated way on K and B and change as B → ∞ .The difference (D1) of the ramification conditions in the two models pre-sumably accounts for much of the mismatch. The S n -number fields detectedby the random polynomial model are always unramified in the field sense,but the random monic polynomial models do not detect some S n -number fields not ramified at ( p ). We should really replace the p -part of Disc( f ) withthe p -part of D K , with K = Q ( θ ), which involves studying the p -adic coeffi-cients of f ( x ). The model of Bhargava is based on a mass formula counting p -adic ´etale extensions with weights, and the weights matter. However fromthe viewpoint of the difference (D2) it is not immediately clear that suchweighted sums will conspire to produce the nice limiting values given in The-orem 2.4. To understand difference (D2) better it might be interesting tostudy an auxiliary question: for each pair of S n -number fields K , K whatis the behavior as B → ∞ of the ratio of the number of f ( x ) in the box ofsize B that generate the field K (resp. K ) and satisfy p ∤ Disc( f ( x )). Doesthis quantity have a limiting value and if so, how does it depend on K and K ?We conclude that there are observable structural differences between thetwo models. We do not currently have a conceptual explanation how thesestructural differences account for and quantitatively explain the differencesin the limiting densities of the two models.4. Splitting Measures
In this section we define and study the one-parameter family of splittingmeasures ν ⋆n,z ( z ∈ C ) on the symmetric group S n , for each n . We relatethis measure at z = q = p k to finite field factorization of degree n monicpolynomials over F q .4.1. Necklace polynomials.
The number of monic irreducible polynomi-als of degree m over finite fields F q for q = p f are well known to be interpo-latable by universal polynomial M m ( X ) evaluated at value X = q . Recallthat for m ≥ necklace polynomial of degree m is M m ( X ) ∈ Q [ X ] by M m ( X ) := 1 m X d | m µ ( md ) X d = 1 m X d | m µ ( d ) X md , (4.1)where µ ( d ) is the M¨obius function. For m = 0 we set M ( X ) = 1. We notethat M ( X ) = X and M ( X ) = X ( X − M m ( X ) ∈ m Z [ X ] , for m ≥
1. The name “necklace polynomial” was proposed by Metropolis andRota [35], because the value M m ( k ) for positive integer k has a combinatorialinterpretation as counting the number of necklaces of m distinct coloredbeads formed using k colors which have the property of being primitive inthe sense that their cyclic rotations are distinct (Moreau [36]). In 1937 Witt[46, Satz 3] showed that M m ( k ) counts the number of basic commutatorsof degree m in the free Lie algebra on k generators. See the discussion inHazewinkel [23, Sect. 17].For later use we give some basic properties of M m ( X ). PLITTING BEHAVIOR OF S n -POLYNOMIALS 15 Lemma 4.1. (1) Let q = p k be a prime power and let N irredm ( F q ) count thenumber of irreducible monic polynomials in F q [ X ] of degree m . Then M m ( q ) = N irredm ( F q ) . (2) The polynomial M m ( X ) ∈ Q [ X ] is an integer-valued polynomial, i.e.one has M m ( k ) ∈ Z for all k ∈ Z .Proof. (1) The well known formula N irredm ( F q ) = M m ( q ) was found byGauss in the unpublished Section 8 of Disquisitiones Arithmeticae,
Articles342 to 347, see Gauss [21, pp. 212–240], cf. Maser [34]. A proof is given inRosen [39, p. 13].(2) To verify that a polynomial in m Z [ X ] is integer-valued, it sufficesto check the integrality property holds at m consecutive integer values of X . The integrality property at positive integers follows from the countinginterpretation of the values M m ( j ) of Moreau [36], see also [35]. (cid:3) We next obtain bounds on the size of M m ( X ) which will be used inSections 4.5 and 4.6 to establish non-negativity properties of the z -splittingdistributions for certain parameter ranges. Lemma 4.2. (1) The necklace polynomial M m ( X ) has M m (0) = 0 for m ≥ . In addition M m (1) = for m = 1 , for m ≥ . One has ( X − ∤ M m ( X ) for all m ≥ .(2) One has M m ( t ) > , for all real t ≥ . In addition, for ≤ j ≤ m there holds for real t > m − , M j ( t ) > (cid:22) mj (cid:23) − . (4.2) (3) For m ≥ one has ( − m M m ( − t ) > , for all real t ≥ . (4.3) In addition, for each m ≥ and t > with t ( t + 1) > m − , there holds for ≤ j ≤ m/ , M j ( t ) > (cid:22) m j (cid:23) − . (4.4) Proof. (1) We have M m (0) = 0 since it has no constant term for m ≥
1. For m ≥ M m (1) = P d | m µ ( d ) , which yields M m (1) = 0 for m ≥ Gauss found this formula on August 25, 1797, according to his
Tagebuch , see Frei [17].
Thus X ( X − | M m ( X ) for m ≥
2. The relation ( X − does not divide M m ( X ) follows from M ′ m ( X ) | X =1 = 1 m (cid:16) X d | m µ ( d ) md (cid:17) = Y p | m (cid:16) − p (cid:17) > . (2) For m ≥ t ≥
2, one has m M m ( t ) ≥ t m − ⌊ m/ ⌋ X j =1 t j = t m − t m +1 − tt − ! ≥ t m − t m +1 + t > . (4.5)For the second part, suppose m ≥ ≤ j ≤ m . We have for j = 1 and t > m − M ( t ) = t > m − . For j = 2 and t > m − M ( t ) = 12 t ( t − ≥ j m k − , the last inequality being immediate for m = 2 and easy for m ≥ . Finally,for 3 ≤ j ≤ m , and t > m −
1, we have t j − t j +1 ≥ , whence by (4.5), jM j ( t ) ≥ t > m which gives (4.2) in this case.(3) To establish M m ( − t ) > t >
2, note that for m = 1 one has − M ( − t ) = t > t >
0. For m ≥ t > m M m ( − t ) ≥ t m − ( ⌊ m/ ⌋ X j =1 t j ) ≥ t m − t m + t > . (4.6)For the second part, it suffices to show for 1 ≤ j ≤ m/ j ) M j ( − t ) > m − j (cid:16) ≥ j ( (cid:22) m j (cid:23) − (cid:17) . For 2 j = 2 we have by hypothesis2 M ( − t ) = t ( t + 1) > m − . For 2 j ≥ m ≥
6, the condition t ( t + 1) > m − t >
2. Then(4.6) applies and we obtain.2 jM j ( − t ) ≥ t j − t j + t ≥ t ( t + 1) > m − > m − j. as required. The remaining case is m = 4 and 2 j = 4, where m − j = 0,the condition t ( t + 1) > t >
1, whence4 M ( − t ) = t − t > , as required. (cid:3) PLITTING BEHAVIOR OF S n -POLYNOMIALS 17 Cycle polynomials.
To any partition µ ⊢ n we associate the cyclepolynomial N µ ( X ) := n Y i =1 (cid:16) M i ( X ) c i ( µ ) (cid:17) (4.7)Here N µ ( X ) ∈ Q [ X ] is a polynomial of degree n (since P ni =1 ic i = n ).The values N µ ( X ) for prime powers X = q = p f count the number ofsquare-free polynomial factorizations of type µ in F q [ X ], as shown in Section4.4. Lemma 4.3. (Properties of Cycle Polynomials)
Let n ≥ . For any parti-tion µ ⊢ n the cycle polynomial N µ ( X ) has the following properties.(1) The polynomial N µ ( X ) ∈ n ! Z [ X ] is integer-valued.(2) The polynomial N µ ( X ) has lead term n Y i =1 i c i ( µ ) c i ( µ )! ! X n = | C µ | n ! X n . (3) The polynomial N µ ( X ) is divisible by X m , where m ≥ counts thenumber of distinct cycle lengths appearing in µ .(4) There holds X µ ⊢ n N µ ( X ) = X n − ( X − . (4.8) Proof. (1) The definition (4.7) implies that N µ ( X ) ∈ d ( µ ) Z [ X ] with d ( µ ) = n Y i =1 i c i ( µ ) c i ( µ )!By comparison with equation (2.2) we have d ( µ ) = n ! | C µ | , which shows that d ( µ ) divides n !, with equality when µ = h n i . The integrality of N µ ( k ) for k ∈ Z follows from the definition using the integrality of all M i ( k ) (Lemma4.1(2)).(2) The property follows by direct calculation of the top degree term in(4.7).(3) The divisibility property is immediate from the definition (4.7) since X divides (cid:16) M i ( X ) c i ( µ ) (cid:17) whenever c i ( µ ) > n , so itsuffices to verify that the identity holds at n + 1 distinct values of X . To thisend, we make use of a combinatorial interpretation of N µ ( X ) for X = q = p k a prime power, given in Proposition 4.5 below. The sum on the left evaluatedat X = p k counts all possible degree n monic polynomials over F q [ X ] for q = p k that have a square-free factorization, i.e. nonvanishing discriminantover F q . The resulting polynomial F ( X ) satisfies F ( q ) = q n − q n − , accordingto Proposition 4.5 (1), verifying the identity at X = q . (cid:3) Splitting Measures.
For each n ≥ splitting measures ν ∗ n,z on the symmetric group S n , with familyparameter z ∈ C , by means of their values on conjugacy classes ν ∗ n,z ( C µ ) := 1 z n − ( z − N µ ( z ) = 1 z n − ( z − n Y i =1 (cid:18) M i ( z ) c i ( µ ) (cid:19) . (4.9)For any element g ∈ C µ we set ν ∗ n,z ( g ) := 1 | C µ | ν ∗ n,z ( C µ ) . (4.10)The latter formula coincides with the definition (2.1) for ν ∗ n,z ( g ). Since thisformula is a rational function of z for each µ , with possible poles only at z = 0 ,
1, this defines a complex-valued function on S n constant on conjugacyclasses, for all z on the Riemann sphere b C := C ∪ {∞} except possibly at z = 0 ,
1. The measure at z = ∞ is the uniform measure, ν n, ∞ ( g ) = n ! , aresult that follows from Lemma 4.2(ii)–see also the proof of Theorem 2.3 (3)below. The measure at z = 1 also turns out to be well-defined but is now asigned measure. It is studied by the first author in [28].We next show that these measures have total (complex-valued) mass one. Proposition 4.4.
For n ≥ , for all z ∈ b C r { } and denoting conjugacyclasses in S n by C µ with µ ⊢ n , X µ ⊢ n ν ∗ n,z ( C µ ) = 1 . Equivalently, for all g ∈ S n , X g ∈ S n ν ∗ n,z ( g ) = 1 . Proof.
For z ∈ b C r { , , ∞} the lemma follows from the normalizationidentity (4.8) for the cycle polynomials. It extends by analytic continuationin z to the values z = 1 , ∞ . (cid:3) Splitting measures and finite field factorizations.
A main ratio-nale for the study of z -splitting measures is that when z = q = p k is a primepower these measures occur in the statistics of factorization of monic polyno-mials of degree n in F q [ X ], drawn from a uniform distribution, conditionedon being square-free.Recall that a monic polynomial f ( x ) ∈ F q [ x ] factors uniquely as f ( x ) = Q ki =1 g i ( x ) e i , where the e i are positive integers and the g i ( x ) are distinct,monic, irreducible, and non-constant. We have the following basic factsabout square-free factorizations. Proposition 4.5.
Fix a prime p ≥ , and let q = p f . Consider the set F n,q of all monic polynomials in F q [ X ] of degree n , so that |F n,q | = q n . PLITTING BEHAVIOR OF S n -POLYNOMIALS 19 (1) Exactly q n − of these polynomials have discriminant Disc( f ) = 0 in F q . Equivalently, exactly q n − of these polynomials are not square-free whenfactored into irreducible factors over F q [ X ] .(2) The number N ( µ ; q ) of f ( x ) ∈ F n,q whose factorization over F q intoirreducible factors is square-free of degree type µ := ( µ , ..., µ r ) , with µ ≥ µ · · · ≥ µ r having c i = c i ( µ ) factors of degree i satisfies N ( µ ; q ) = n Y i =1 (cid:16) M i ( q ) c i ( µ ) (cid:17) = N µ ( q ) , (4.11) in which N µ ( X ) denotes the cycle polynomial for µ .Proof. (1) This result can be found in [39, Prop. 2.3]. Another proof, dueto M. Zieve, is given in [45, Lemma 4.1].(2) This result is well known, see for example S. D. Cohen [10, p. 256].It follows from counting all unique factorizations of the given type. (cid:3) This proposition has the following consequence.
Proposition 4.6.
Consider a random monic polynomial g ( X ) of degree n drawn from F q [ x ] with the uniform distribution, where q = p f . Then theprobability of g ( x ) having a factorization into irreducible factors of split-ting type µ ∈ T ∗ n , conditioned on g ( x ) having a square-free factorization, isexactly ν ∗ n,q ( C µ ) . That is, ν ∗ n,q ( C µ ) = Prob[ g ( x ) has splitting type µ | g ( x ) is square-free ] . Proof.
Proposition 4.5 (1), and (2) together evaluate the conditional prob-abilityProb[ g ( x ) has splitting type C µ | g ( x ) is square-free] = 1 q n − q n − n Y i =1 (cid:16) M i ( q ) c i ( µ ) (cid:17) . Comparing the right side with the definition (2.3) of the splitting measureshows that it equals ν ∗ n,q ( C µ ). (cid:3) Nonnegativity conditions for splitting measures: Positive real z . This paper is concerned with the case that z = t is a real number ( t =0 , z and prove Theorem 2.3, whichspecifies several real parameter ranges where these measures are nonnega-tive, and so define probability measures; these parameter values include allinteger values z = m ≥ Proof of Theorem 2.3.
To decide on nonnegativity or positivity of ν ∗ n,t ( C µ ),it suffices to study the individual terms (cid:0) M i ( t ) c i ( µ ) (cid:1) for 1 ≤ i ≤ n and to shownonnegativity (resp. positivity) of each of the numerators( M i ( t )) c i ( µ ) = M i ( t )( M i ( t ) − · · · ( M i ( t ) − c i ( µ ) + 1) . (4.12) (1) We verify that for t > n − µ , all terms in the definition (4.9)of ν ∗ n,t ( C µ ) for 1 ≤ i ≤ n have (cid:0) M i ( t ) c i ( µ ) (cid:1) > . The positivity of the terms in(4.12) is immediate for n = 1 so suppose n ≥
2. Using Lemma 4.2 (2) for t > n − M i ( t ) > j ni k − ≥ c i ( µ ) − , whence all factors in the product (4.12) are positive, as asserted.(2) For each integer 2 ≤ k ≤ n −
1, the normalizing factor k n − ( k − in thedefinition is positive. Since M i ( X ) is an integer-valued polynomial for all i ≥
1, each term in the product definition of ν ∗ n,k ( C µ ) is a binomial coefficient,hence is nonnegative. This proves nonnegativity of the k -splitting measure.Finally, for the identity conjugacy class C h n i = { e } , for 2 ≤ k ≤ n − ν ∗ n,k ( C h n i ) = 0 , since in these case the i = 1 factor in (4.12) has( M ( k )) n = 0.(3) The limit as t → ∞ is driven by the lead term asymptotics of thepolynomial M i ( t ) . Using P i ic i ( µ ) = n and M i ( t ) = i t i + O ( t i − ) we obtainlim t →∞ ν ∗ n,t ( C µ ) = lim t →∞ n Y i =1 M i ( t ) c i ( µ ) c i ( µ )! t ic i ( µ ) = n Y i =1 c i ( µ )! i c i ( µ ) = | C µ | n ! , as asserted. (cid:3) Nonnegativity conditions for splitting measures: negative real z . We prove complementary results specifying some negative real parame-ter values z = − t < µ n, − t ( C µ ) is nonnegative and so defines aprobability measure. Theorem 4.7.
Let n ≥ . Then for real z = − t < the signed measures ν ∗ n,t on S n have the following properties:(1) For all real values t > having t ( t + 1) > n − , the measure ν ∗ n, − t on S n is strictly positive, so that it defines a probability measure on S n withfull support.(2) For all integers k ≥ having t ( t + 1) ≤ n − the measure ν n, − k isnonnegative and defines a probability measure on S n . This measure does nothave full support. it is zero on the conjugacy class C µ with µ = h n/ i if n is even, and on the conjugacy class with µ = h , ( n − / i if n is odd.(3) There holds for all g ∈ S n , lim t →∞ ν ∗ n, − t ( g ) = 1 n ! . Proof. (1) To show positivity of the measure we keep track of the signs ofall the factors in the definition (4.9). Since t > (cid:0) − t ) n − ( − t − (cid:1) = ( − n . PLITTING BEHAVIOR OF S n -POLYNOMIALS 21 Lemma 4.2 (3) then gives for t ( t + 1) > n − M j ( − t ) > (cid:22) t j (cid:23) − . Since c j ( µ ) ≤ j nj k , we obtain the positivity of all even degree terms, as( M j ( − t )) c j ( µ ) = M j ( − t )( M j ( − t ) − · · · ( M j ( − t ) − c j ( µ ) + 1) > . We assert that all odd degree terms have M j +1 ( − t ) <
0. Assuming this isproved, we obtain Sign(( M j +1 ( − t )) c j +1 ( µ ) ) = ( − c j +1 ( µ ) = ( − (2 j +1) c j +1 ( µ ) .It follows thatSign( ν ∗ n,t ( C µ )) = ( − n ( − P i jc i ( µ ) = ( − n = 1 , showing the required positivity.It remains to show that all M j +1 ( − t ) <
0. This holds for t ≥ t ≥ n ≥
8. For the remaining cases wecheck M ( − t ) = − t < t >
0, and that for 2 j +1 = 3 , , M j +1 ( − t ) = − t j +1 + t < t > ν ∗ n, − k for those positive integer k with k ( k − ≤ n −
2, the argument of (1) still applies with the followingchanges. For even indices 2 j , we use the fact that M j ( − k ) is a positiveinteger, so either the descending factorial remains positive or else is zero ifa zero is encountered. So the sign of this term may be treated as positive.For the odd indices 2 j + 1, either the initial value M j +1 ( − k ) = 0, in whichcase the measure is 0, or else M j +1 ( − k ) < M j +1 ( − k ) < k ≤ M j +1 ( − n = 2 ℓ that for µ = h ℓ i , one has ν ∗ n, − k ( C µ ) = 0 for all positive integers k with k ( k + 1) ≤ n −
2. Here m ( µ ) = ℓ and the integer1 ≤ M ( − k ) = 12 k ( k + 1) ≤ n −
22 = ℓ − , so that the descending factorial ( M ( − m )) ℓ = 0 . One verifies similarly thatfor n = 2 ℓ + 1 ≥ ν n, − k ( C µ ) = 0 for µ = h , ℓ i , where again m ( µ ) = ℓ and n − = ℓ − . (3) This limit behavior follows similarly to the case of Theorem 2.3 (3). (cid:3) Counting S n -Polynomials with Specified Splitting Types Counting monic S n -polynomials with coefficients in a box. Itis well-known that, in a suitable sense, almost all monic polynomials with Z coefficients have a splitting field that is an S n -extension of Q . This wasproved in 1936 by van der Waerden [44], who showed that the fraction of allmonic degree n polynomials in Z [ x ] having all coefficients in a box | a i | ≤ B that have a splitting field with Galois group S n approaches 1 as B → ∞ . Animproved quantitative form of this assertion was given in 1973 by Gallagher[20], which we formulate as follows. Theorem 5.1 (Gallagher) . For integer B ≥ let F n ( B ) be the set of monic,degree n polynomials in Z [ x ] with all coefficients in the box [ − B +1 , B ] ; thereare (2 B ) n such polynomials. Let E n ( B ) denote the proportion of polynomialsin F n ( B ) which do not have splitting field with Galois group S n . Then thereexists a positive constant α n , depending only on n , such that for all B > , E n ( B )(2 B ) n ≤ α n log B √ B . (5.1)We remark that all polynomials with coefficients in the box F n ( B ) satisfy | Disc( f ) | ≤ (4 B ) n ( n − . (5.2)Indeed, we have Disc( f ) = Q ≤ i Let n ≥ be given, and let S = { p , ..., p r } be a finite set ofprimes and let U = { µ , ..., µ r } be a corresponding set of splitting symbols.(1) Let F n ( B ; S ) denote the set of all polynomials f ( x ) in F n ( B ) suchthat gcd(Disc( f ) , Q ri =1 p i ) = 1 . Then lim B →∞ { f ( x ) ∈ F n ( B ; S ) } { f ( x ) ∈ F n ( B ) } = r Y i =1 (cid:16) − p i (cid:17) . (5.4) (2) Let F n ( B : {S ; U } ) denote the set of all f ( x ) in F n ( B, S ) such that: PLITTING BEHAVIOR OF S n -POLYNOMIALS 23 (i) f ( x ) has splitting field K f that is an S n -extension of Q . (ii) The splitting type of f ( x ) (mod p i ) is C µ i for ≤ i ≤ r. Then lim B →∞ { f ( x ) ∈ F n ( B ; {S , U } ) } { f ( x ) ∈ F n ( B ; S ) } = r Y i =1 ν ∗ n,p i ( C µ i ) . (5.5)We note that the condition gcd(Disc( f ) , Q ri =1 p i ) = 1 on a monic irre-ducible polynomial guarantees that the field K = Q ( θ ) generated by a singleroot of f ( x ) is unramified over all the primes in S . In that case, the dis-criminant Disc ( f ) detects the discriminant of the ring O f = Z [1 , θ, ..., θ n − ] , which is a subring of the full ring of integers O ( K ) of the field K = Q ( θ )generated by a root of the polynomial. We haveDisc( f ) = Disc( K )[ O ( K ) : O f ] , so that p ∤ Disc( f ) implies p ∤ Disc( K ).We will derive Theorem 5.2 from two quantitative estimates given below.We begin with an estimate for the event gcd(Disc( f ) , Q ri =1 p i ) = 1 . Lemma 5.3. Let n ≥ . Let S = { p , p , ..., p r } and M = Q ri =1 p i . Thenfor B ≥ nM , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { f ( x ) ∈ F n ( B ; S ) } { f ( x ) ∈ F n ( B ) } − r Y i =1 (1 − p i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ nMB . Proof. For each prime p the behavior of Disc ( f ) (mod p ) is determined by( a , a , ..., a n − ) (mod p ) . Thus if M divides B then Proposition 4.5 (1)shows that exactly a fraction of p of these polynomials have Disc( f ) ≡ p ) . The polynomials are labelled by lattice points in the closedbox [ − B + 1 , B ] n , and we call a lattice point admissible if it correspondsto a polynomial in F n ( B, S ). For a general B we first round down to abox of side B ′ = M (cid:4) BM (cid:5) , and using there the Chinese remainder theoremwe find exactly (2 B ′ ) n Q ri =1 (1 − p i ) admissible polynomials in the smallerbox belong to F n ( B, S ). This number undercounts (2 B ) n Q ri =1 (1 − p i ) byamount Q ri =1 (1 − p i )((2 B ) n − (2 B ′ ) n ). Similarly we may round up to a boxof side B ′′ = M ⌈ BM ⌉ and using there a similar argument we find exactly Q ri =1 (1 − p i )(2 B ′′ ) n admissible polynomials in the larger box. Thus(2 B ′ ) n r Y i =1 (cid:16) − p i (cid:17) ≤ |F n ( B ; S ) | ≤ (2 B ′′ ) n r Y i =1 (cid:16) − p i (cid:17) We now use the inequality, valid for real | x | ≤ n ,1 + 2 n | x | ≥ (1 + x ) n ≥ − n | x | . Since B ′′ − B ′ ≤ M , the inequality gives for B ≥ nM ,(2 B ′′ ) n − (2 B ′ ) n ≤ (2 B ) n (cid:16) (1+2 n B ′′ − BB ) − (1 − n B − B ′ B ) (cid:17) ≤ (2 B ) n ( 2 nMB ) . (5.6)This yields the estimate { f ( x ) ∈ F n ( B ; S ) } = (1 + ǫ n ( B ; S ))(2 B ) n r Y i =1 (1 − p i ) , (5.7)with | ǫ n ( B ; S ) | ≤ nMB . Dividing both sides by { f ( x ) ∈ F n ( B ) } = (2 B ) n yields the desired bound. (cid:3) Now we derive the main estimate from which Theorem 5.2 will follow. Theorem 5.4. Let n ≥ . Let S := { p , p , ..., p r } be a finite set of primesand let U := { µ , ..., µ r } be a set of splitting types. Let F n ( B : { S ; U } ) denotethe set of all polynomials f ( x ) in F n ( B ) such that: (i) gcd(Disc( f ) , Q ri =1 p i ) = 1 ; (ii) The splitting type of f ( x ) (mod p j ) is C µ j , for ≤ j ≤ r ;(iii) f ( x ) has splitting field K f that is an S n -extension of Q .Then, setting M = Q i p i , for B ≥ nM there holds | { f ( x ) ∈ F n ( B ; {S , U } ) } { f ( x ) ∈ F n ( B ; S ) } − r Y i =1 ν ∗ n,p i ( C µ i ) | ≤ r Y i =1 (1 − p i ) − α n log B √ B + 4 nMB , Proof. Let F n ( B, {S , U } ) + denote the set of all polynomials f ( x ) in F n ( B )that satisfy conditions (i) and (ii) above. Theorem 5.1 then gives0 ≤ |F n ( B, {S , U } ) + | − |F n ( B, {S , U } ) | ≤ (2 B ) n (cid:16) α n log B √ B (cid:17) . For splitting types on box of side B ′ = M (cid:4) BM (cid:5) by reduction (mod M )together with Proposition 4.5 (2) and the Chinese remainder theorem weget a product distribution of all splitting types (mod p i ) for 1 ≤ i ≤ r , |F n ( B ′ , {S , U } ) + | = (2 B ′ ) n n Y i =1 p ni N µ i ( p i ) , where N µ i ( · ) is a cycle polynomial. We have a similar formula for an en-closing box of side B ′′ = M ⌈ BM ⌉ , with (2 B ′′ ) n replacing (2 B ′ ) n . Assuming B ≥ nM we obtain by an application of (5.6) that |F n ( B, {S , U } ) + | = (1 + ǫ n ( B ; {S , U } ))(2 B ) n n Y i =1 p ni N µ i ( p i ) , PLITTING BEHAVIOR OF S n -POLYNOMIALS 25 with the error estimate | ǫ n ( B, {S , U } ) | ≤ nMB . Next we note that q n N µ ( q ) = (cid:16) − q (cid:17) ν ∗ n,q ( C µ ) . Substituting this for each p i in the formula above and using our original bound for |F n ( B, {S , U } ) | yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) |F n ( B, {S , U } ) | − (2 B ) n n Y i =1 (cid:16) − p i (cid:17) ν ∗ n,p i ( C µ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (2 B ) n (cid:16) nMB n Y i =1 (cid:16) − p i (cid:17) + α n log B √ B (cid:17) . For B ≥ nM , we replace (2 B ) n Q i (1 − p i ) with |F n ( B ; S ) | using (5.7) weobtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) |F n ( B ; {S , U } ) | − |F n ( B ; S ) | r Y i =1 ν ∗ n,p i ( C µ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (2 B ) n (cid:16) nMB n Y i =1 (cid:16) − p i (cid:17) + α n log B √ B (cid:17) The result follows on dividing both sides by |F n ( B ; S ) | = (1 − ǫ n ( B ; S )(2 B ) n n Y i =1 (cid:16) − p i (cid:17) , noting for B ≥ nM that (5.7) implies | ǫ n ( B : S ) | ≤ . (cid:3) Proof of Theorem 5.2. This follows directly from Lemma 5.3 and Theo-rem 5.4 on letting B → ∞ . (cid:3) Remark. The conclusion in Theorem 5.2 is insensitive to the shape of thebox bounding the coefficients as long as the box increases homotheticallyas B → ∞ , e.g. one can use − c n,j B < a j < c n,j B , where c n,j are positiveconstants independent of B , and derive exactly the same limiting formula.For example, c n,j = (cid:16) nj (cid:17) is another natural choice.5.3. Existence of S n -number fields with specified splitting types:Proof of Theorem 2.5. We first remark on a special property of the sym-metric group S n as a Galois group, represented as a permutation groupacting transitively on the roots of a polynomial, that distinguishes it fromsome of its subgroups. Let G be a permutation group G ⊂ S n (i.e. apermutation representation of the abstract group G ). The elements in aconjugacy class in G necessarily have the same cycle type as permuta-tions, but the converse need not hold. That is, the cycle type of a con-jugacy class in G need not determine it uniquely. For example, the group G = { (12)(34) , (13)(24) , (14)(23) , (1)(2)(3)(4) } ⊂ S is abelian so all con-jugacy classes have size 1 but three of these classes have identical cyclestructures. This uniqueness property does hold for cycle types for the fullsymmetric group S n , which has the consequence that the cycle type of an S n polynomial having a square-free factorization ( mod p ) uniquely determines the Artin symbol for an S n -number field obtained by adjoining one root ofit. Proof of Theorem 2.5. (1) ⇒ (3). By hypothesis we are given S = { p , ..., p r } and splitting types U = { µ , .., µ r } with the property that all ν ∗ n,p i ( C µ i ) > S n -number fields K whose Galois closure K ′ has the given Artin symbols h K ′ / Q ( p i ) i = C µ i , ≤ i ≤ r (5.8)is infinite, by showing it is arbitrarily large. Since the splitting type of apolynomial f ( X ) generating an S n -number field modulo p determines thecorresponding Artin symbol, it suffices to specify factorizations of polyno-mials (mod p i ) which we can do using Theorem 5.2.Given k ≥ S k := S S S ∗ k with S ∗ k = { p r +1 , · · · , p r + k } beinga set of k auxiliary primes that satisfy n ≤ p r +1 < p r +2 < ... < p r + k and disjoint from the primes in S . In that case we may choose splittingsymbols U k := { µ r +1 , ..., µ r + k } arbitrarily in S n for the auxiliary primes andthe condition ν ∗ n,p r + j ( C µ r + j ) > p i guaranteesthat the polynomial discriminant is relatively prime to p p · · · p r + k and thisproperty guarantees that (P1) holds. Theorem 5.2 now implies the existenceof infinitely many S n -polynomials having the given splitting behavior at all r + k primes; thus (P2) holds for such fields. In particular there exists atleast one such S n -number field K exhibiting the given splitting behavior.Since each S n for n ≥ k different S n -number fields, all of which matchthe splitting types C µ i for 1 ≤ i ≤ r in (5.8) and which are distinguishableamong themselves by how the auxiliary primes p r + j , 1 ≤ j ≤ k split. Since k can be arbitrarily large, the result follows.(3) ⇒ (2). Immediate.(2) ⇒ (1). By hypothesis the given field K possesses is a monogenic or-der Z [1 , θ, ..., θ n − ] satisfying (P1). The minimal polynomial for θ is thena monic polynomial f ( x ) ∈ Z [ X ] which satisfies gcd( Disc ( f ) , p · · · p r ) = 1.This polynomial then has square-free factorization (mod p i ) yielding thesplitting types C µ i for 1 ≤ i ≤ r , see Lang [30, I. § 8, Proposition 25].We next observe that the splitting type conditions are congruence condi-tions ( mod p · · · p n ) on the coefficients of f , and they enforce the conditiongcd( Disc ( f ) , p p · · · p r ) = 1. These congruence conditions are satisfied fora positive proportion of polynomials in the box, so in Theorem 5.2 the leftside of (5.5) is positive, which certifies that each ν ∗ n,p i ( C µ i ) > (cid:3) The hypothesis of Lang’s Proposition 25 requires Z [1 , θ, · · · , θ n − ] to be integrallyclosed, i.e. the full ring of integers O k . As he notes, the argument can be done bylocalizing over each prime ideal ( p i ), and here ( Disc ( f ) , p i ) = 1 implies that the integralclosure condition holds locally. PLITTING BEHAVIOR OF S n -POLYNOMIALS 27 Vanishing values of splitting measures: Proof of Theorem 2.6. We characterize pairs ( n, p, µ ) where ν ∗ n,p ( C µ ) = 0. Proof of Theorem 2.6. ( C ⇔ ( C p = 0 , ν ∗ n,p ( C µ ) =0 holds if and only if N µ ( p ) = 0 . By Proposition 4.5 (2) the latter con-dition holds if any only if no degree n monic polynomial in F p [ X ] with Disc ( f ) = 0 ∈ F p has a square-free factorization of splitting type µ . Thelatter condition is exactly (C2).( C ⇔ ( C S n -number field K which at ( p ) is unram-ified and has splitting type µ . Now the equivalence of Theorem 2.6 appliedfor a single prime p = p shows that ν ∗ n,p ( µ ) > , which is equivalent to thecondition that ( C 1) does not hold.We remark that this argument does not establish whether or not thereexist any S n extensions K which satisfy condition (C3) for given splittingdata µ . (cid:3) Number of S n -Polynomials with Specified Splitting Typesover Number Fields We consider polynomials with coefficients drawn from an algebraic num-ber field k , not necessarily Galois over Q . We set [ k : Q ] = d , and say thatan extension L/k with [ L : k ] = n is a relative S n -number field if the Galoisclosure L ′ of L over k has Gal ( L ′ /k ) ≃ S n . We let D k denote the absolutediscriminant of k over Q .Let O k denote the ring of algebraic integers in k . We consider monicpolynomials f ( x ) = x n + n − X j =0 α j x j , with all α j ∈ O k . Choose an integral basis O k = Z [ ω , ω , ..., ω d ], and letΩ = ( ω , ..., ω d ) denote this (ordered) integral basis. We now have α j = d X k =1 m j,k ω k , ≤ j ≤ n, for unique m i,j ∈ Z . We define F n ( B ; Ω) to be the set of all monic degree n polynomials over O k whose coefficients have all m i,j satisfying − B + 1 ≤ m i,j ≤ B , so there are (2 B ) nd polynomials in the box.Next we let S = { p , ..., p r } denote a finite ordered set of (distinct) primeideals in O k . We allow different ideals in the list to have residue class fieldsof the same characteristic, i.e. to lie over the same rational prime. We set N k/ Q p j = p f j j . We let U = { µ , ..., µ r } denote a finite ordered set of splittingtypes of S n (the different µ j need not be distinct). Theorem 6.1. Suppose that k/ Q is a number field, not necessarily Galoisover Q , Let S = { p , ..., p r } be an ordered finite set of distinct prime idealsin O k and let F q i denote the residue class field for p i , with q i = N p i = p f i i .Suppose U = { µ , ..., µ r } is a given ordered set of splitting symbols. Thenfor fixed n ≥ , the following hold.(1) Let F n ( B ; S , Ω) denote the set of all degree n polynomials f ( x ) in F n ( B ; Ω) such that gcd( Disc ( f ) , Q ri =1 p i ) = (1) , viewed as ideals in O k .Then lim B →∞ { f ( x ) ∈ F n ( B ; S , Ω) } { f ( x ) ∈ F n ( B ; Ω) } = r Y i =1 (cid:16) − q i (cid:17) . (6.1) (2) Let F n ( B : {S ; U } , Ω) denote the set of all f ( x ) in F n ( B ; S , Ω) suchthat: (i) The splitting type of f ( x ) (mod p i ) is C µ i for ≤ i ≤ r. (ii) f ( x ) has relative splitting field K f over k that is an S n -extension of k .Then lim B →∞ { f ( x ) ∈ F n ( B ; {S , U } ) , Ω } { f ( x ) ∈ F n ( B ; S , Ω) } = r Y i =1 ν ∗ n,q i ( C µ i ) . (6.2) Proof. This result parallels the proof of Theorem 5.2. We only sketch thedetails, indicating the main changes needed. Suppose [ k : Q ] = d .Firstly, we have { f ( x ) ∈ F n ( B ; Ω) } = (2 B ) nd . The condition for the polynomial discriminant gcd( Disc ( f ) , Q ri =1 p i ) = (1)is exactly that the polynomial f ( x ) have square-free factorization ( mod p i )for 1 ≤ i ≤ r . Set M = Q ri =1 q i = Q ri =1 ( p i ) f i . For the limit in (1) we obtainan exact count when going through boxes having all sides B = M m for someinteger m ≥ 1, which is |F n ( B ; S , Ω }| = (2 B ) nd r Y i =1 (cid:16) − N p i (cid:17) = (2 B ) nd r Y i =1 (cid:16) − q i (cid:17) For each prime ideal p i this holds using Proposition 4.5 (1) since we have anintegral multiple of complete residue systems (mod p i ) in the box, and itholds for all p i simultaneously using the Chinese remainder theorem for thebox. Allowing a general B adjusts this formula by a multiplicative amount1 + O ( ndMB ), and letting B → ∞ yields (6.1).Secondly, we introduce F n ( B ; {S , U } , Ω) + to be those elements of F n ( B ; S , Ω)that satisfy condition (i) only. We then have a bound for the number of these f ( x ) that do not give S n -extensions of k , which is0 ≤ F n ( B ; {S , U } , Ω) + − F n ( B ; {S , U } , Ω) ≤ α n ( k )(2 B ) nd d log B √ B d . PLITTING BEHAVIOR OF S n -POLYNOMIALS 29 This result follows using an upper bound of Cohen [11, Theorem 2.1], in hisresult specifying that F t ( x ) = X n + P n − i =0 t i X i , that K = k , and noting theGalois group G = S n for F t ( X ) over the function field k ( t , · · · , t n ) . Thirdly, on restricting the box size to the special form B = M m with m ≥ 1, one gets an exact count |F n ( B ; {S , U } , Ω }| = (2 B ) nd r Y i =1 N p i ) n N µ i ( N p i ) . This formula is equivalent to |F n ( B ; {S , U } , Ω }| = (2 B ) nd r Y i =1 (cid:16) − q i (cid:17) ν ∗ n,q i ( C µ i ) . Changing the box size to an arbitrary integer B introduces at most a mul-tiplicative roundoff error of 1 + O ( ndMB ) . Fourthly, we combine the above estimates to obtain an analogue of The-orem 5.4, stating that | |F n ( B ; {S , U } , Ω) ||F n ( B ; S , Ω) | − r Y i =1 ν ∗ n,q i ( C µ i ) | ≤ r Y i =1 (1 − q i ) − α n ( k ) d log B √ B d + 4 ndMB . The formula (6.2) follows on letting B → ∞ . (cid:3) Remark. The conclusion in Theorem 6.1 is insensitive to the shape of the boxbounding the coefficients as long as it is increased homothetically as B → ∞ ,e.g. − c n,j B < a j < c n,j B , where c n,j are positive constants independent of B .We next obtain a result parallel to Theorem 2.5 on the existence of rel-ative S n -number fields K over k having prescribed splitting above a givenfinite set of prime ideals S = { p i : 1 ≤ i ≤ r } , and setting N k/ Q p i = ( q i ),provided that all the quantities ν ∗ n,q i ( C µ i ) > 0. We follow the conventionthat a relative S n -number field K over k is a degree n extension of k whoseGalois closure over k has Galois group S n . We recall that the (relative) dis-criminant Disc ( O K | O k ) of any order O of K that contains O k is that idealof O k that is generated by the discriminants ( α , ..., α n ) of all the bases of K/k which are contained in O . [38, III (2.8)]. The prime ideal powers divid-ing the relative discriminant can be computed locally [37, Prop. 5.7, p. 219]. Theorem 6.2. Let k/ Q be a number field, not necessarily Galois over Q . Let S = { p , ..., p r } denote a finite set of prime ideals of k . and let U = { µ , ..., µ r } with µ j ⊢ n be a prescribed set of splitting symbols for theseprime ideals. Set q i = N k/ Q p i . Then the following conditions are equivalent. (1) The positive measure condition ν ∗ n,q i ( C µ i ) > for ≤ i ≤ r holds. (2) There exists a relative S n -number field K/k having the following twoproperties: (P1- k ) The field K contains a monogenic order O = O k [1 , θ, ...θ n − ] whose relative discriminant Disc ( O | O k ) is relatively prime to p p · · · p r . (P2- k ) The Galois closure K spl of K over k is unramified at all primeideals above those in S and the primes in S have prescribedArtin symbols h K spl /k ( p i ) i = C µ i , ≤ i ≤ r. (3) There exist infinitely many relative S n -number fields K over k havingproperties (P1- k ) and (P2- k ) .Proof. The proof parallels that of Theorem 2.5, using Theorem 6.1 in placeof Theorem 5.2. For (1) ⇒ (3) we use the fact that for a monic polynomial f ( x ) ∈ O k [ x ] that is irreducible over O k one has the equality of polynomialdiscriminants and relative discriminants of the associated monogenic orderin K = k ( θ ), for θ a root of f ( x ). That is, setting O f := O k [1 , θ, · · · , θ n − ].one has the equality ( Disc ( f )) O k = Disc [ O f | O k ] . (6.3)of O k -ideals; here ( Disc ( f )) is a principal ideal. We use this fact to showthat (P1- k ) is satisfied, and apply Theorem 6.1 to show (P2- k ) is satisfied.For (2) ⇒ (1) the hypothesis (P1- k ) with the identity (6.3) implies p i ∤ ( Disc ( f )) as an O k -ideal and the square-free factorization of f ( x ) (mod p i )for each of the p i . This fact gives the required Artin symbols C µ i , andpositive density follows by Theorem 6.1 since all conditions imposed arecongruence conditions. (cid:3) To conclude the paper we formulate a generalization of Theorem 2.6. Fora relative extension K/k of degree n we say that a prime ideal p of O k is called an essential relative discriminant divisor if it divides the relativediscriminants Disc ( O | O k ) all monogenic orders O := O k ]1 , θ, · · · , θ n − ] ofthe field K over k . Theorem 6.3. Let a number field k together with a prime ideal p be given.Let p have ideal norm N k/ Q ( p ) = ( q ) = ( p k ) . For a set of splitting types µ ⊢ n , with n ≥ , the following three conditions are equivalent. (C1- k ) The splitting measure at z = q = p k has ν ∗ n,q ( C µ ) = 0 . (C2- k ) For all degree n monic integer polynomials f ( x ) with coefficientsin O k whose (mod p ) factorization has splitting type µ , the relativediscriminant Disc ( O f | O k ) is divisible by p . PLITTING BEHAVIOR OF S n -POLYNOMIALS 31 (C3- k ) All relative S n -extensions K of k in which p is unramified and hassplitting type µ necessarily have p as an essential relative discrimi-nant divisor.Proof. The proof parallels that of Theorem 2.6. We note only that to es-tablish the equivalence (C1- k ) ⇔ (C2- k ), one uses (6.3). (cid:3) In cases where (C1- k ) holds this proof does not establish that there existany fields satisfying (C3- k ).7. Generalizations Characteristic polynomials of random integer matrices. Theproblem studied in this paper can be viewed as a special case of study ofcharacteristic polynomials of random matrices. One may consider randommatrices drawn from a group like GL ( n, Z ) with constraints on the size ofthe matrix A = [ a i,j ] ( measured in some matrix norm), and also puttingside conditions on the allowed elements. The problem for degree n polyno-mials above can be encoded as such random n × n matrices (having entries | a i,j | ≤ B ) by mapping the polynomial f ( x ) to the companion matrix hav-ing characteristic polynomial f ( x ). After reduction (mod p ) from GL ( n, Z )one obtains a particular distribution of random matrices having entries overthe finite field F p with a side condition forcing many matrix entries to bezero. Our imposed restriction on factorization of polynomials being square-free corresponds requiring that the associated matrices in GL ( n ; F p ) havedistinct eigenvalues, i.e. they belong to semisimple conjugacy classes. Onecan ask whether there are further interesting generalizations of the model ofthis paper results in the random matrix context.There are many results known considering random integer matrices inmore general models. In 2008 Kowalski [27, Chap. 7] showed that thecharacteristic polynomial of a random matrix in SL ( n, Z ) drawn using arandom walk is an S n -polynomial with probability approaching 1 as thenumber of steps increases. For splitting fields of characteristic polynomials ofrandom elements drawn from more general split reductive arithmetic groups G see work of Gorodnik and Nevo [22], Jouve, Kowalski and Zywina [25]. Intheir framework the Galois group S n is replaced by the Weyl group W ( G )of the underlying algebraic group G ; the case W ( G ) = S n corresponds to G = SL n . Lubotzky and Rosenzweig [32] give a further generalization to awider class of groups with coefficients in a wider class of fields, where the“generic” Galois group of a random element may have a more complicatedbehavior.There are also many results known on the distribution of characteristicpolynomials of random matrices over finite fields F q ; this subject is surveyedin Fulman [18]. His paper puts emphasis on M at ( n, F q ) and GL ( n, F q ), andincludes results on factorization type of characteristic polynomials (see also[19]). Example 2 in [18, Section 2.2] observes that the factorization type for a uniformly drawn matrix in M at ( n, F q ) has a distribution dependingon n and q that approaches that of a random degree n monic polynomial in F q [ X ] as q → ∞ . Fulman [18, Section 3.1] also introduces a family of prob-ability measures M GL,u,q on conjugacy classes of GL ( n, F q ), which whenconditioned on fixed n do not depend on the parameter u and have therational function interpolation property in the parameter q . They there-fore extend to a complex parameter z , defining complex-valued measures M GL,z,q . He remarks [18, Section 3.3] that this distribution coincides withthe distribution on partitions describing the Jordan block structure of a ran-dom unipotent element of GL ( n, q ). It would be interesting to determinewhether the measures M GL,u,q have any relation to the splitting measuresstudied in this paper.7.2. Square-free polynomials and homological stability. The split-ting measures ν ∗ n,q ( C µ ) count the relative fraction of monic square-free poly-nomials (mod p ) that have a given factorization type in F q [ x ]. Recently, asa special case of a general theory, Church, Ellenberg and Farb [6] observedthat the monic square-free polynomials in F q [ x ] for q = p k label points inan interesting moduli space Y n ( F q ) defined over F q , the complement of thediscriminant locus, which carries an S n -action. They relate point countson the space Y n ( F q ) specified by factorizations of square-free polynomials in F q [ x ] to the topology of the configuration space X n ( C ) = P Conf n ( C ) := { ( z , z , · · · , z n ) : z i ∈ C , z i = z j } , which itself carries an S n -action. The configuration space PConf n ( C ) isan affine variety which is the complement of a set of hyperplanes. (It is aspecial case of a discriminant variety, see Lehrer [31].) Church, Ellenbergand Farb study the S n -representations produced by the S n -action on thehomology of this space and show certain homological stability properties ofthese representations hold as n → ∞ . They then study limiting behaviors ofpolynomial statistics of these points attached to a fixed multivariate poly-nomial P ( x , ..., x m ) ∈ Q [ x , ..., x m ] and relate this behavior to homologicalstability.The statistics they study over Y n ( F q ) can be expressed in terms of the q -splitting measures ν ∗ n,q ( · ), which may permit an alternative way to viewsome of their results. We hope to consider this topic further elsewhere.For general results on homological stability properties under S n -actionssee Church et al [4], [5], [7]. Acknowledgments. 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