Splitting Behavior of $S_n$-Polynomials

We analyze the probability that, for a fixed finite set of primes S, a random, monic, degree n polynomial f(x) with integer coefficients in a box of side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with splitting field over the rationals having Galois group $S_n$; (ii) the polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii) f(x) has a prescribed splitting type at each prime p in S. The limit probabilities as $B \to \infty$ are described in terms of values of a one-parameter family of measures on $S_n$, called splitting measures, with parameter $z$ evaluated at the primes p in S. We study properties of these measures. We deduce that there exist degree n extensions of the rationals with Galois closure having Galois group $S_n$ with a given finite set of primes S having given Artin symbols, with some restrictions on allowed Artin symbols for p<n. We compare the distributions of these measures with distributions formulated by Bhargava for splitting probabilities for a fixed prime $p$ in such degree $n$ extensions ordered by size of discriminant, conditioned to be relatively prime to $p$.


Introduction
By an S n -polynomial we mean a degree n monic polynomial f (x) with integer coefficients whose splitting field K f /Q obtained adjoining all roots of f (x) has Galois group 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 [43] showed a result implying 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 the cardinality of the exceptional set; see Section 5.1.
This paper considers an extension of this problem which studies the set of polynomials with coefficients in a box [−B, B] n which are S n -polynomials that have prescribed splitting behavior at a given finite set of primes {p k : 1 ≤ k ≤ r}. The frequencies of such polynomials no longer have limiting density 1 as the box size B → ∞. In this paper we determine the limiting splitting densities as B → ∞, after conditioning on the requirement that Date: August 9, 2014August 9, . 1991 Mathematics Subject Classification. Primary 11R09; Secondary 11R32, 12E20, 12E25.
The first author was partially supported by NSF grants DMS-1101373 and DMS-1401224.
the discriminant Disc(f ) of the polynomial f be relatively prime to r i=1 p i . This conditioning restriction imposes a non-ramification condition, requiring the polynomials to have square-free factorizations (mod p). We obtain the limiting splitting densities, which define an interesting set of probability distributions on the symmetric group S n , and study properties of these densities. These densities are inherited from corresponding densities for splitting of polynomials in p-adic fields recently studied by the second author ( [44]), which in turn arise from splitting probabilities for polynomials over finite fields.
1.1. Existence and properties of splitting density measures. The splitting types of a polynomial (mod p) are described by partitions µ of n, which are identified with conjugacy classes on the symmetric group S n . For each n ≥ 1, and for each prime p in Section 2.1 we define splitting measures ν * n,p (·) on S n which are constant on conjugacy classes C µ of S n . We show the following results, whose precise statements are given in Section 2.
(i) The limiting probabilities as B → ∞ at p for degree n monic polynomials f (x) conditioned on p ∤ Disc(f ) to have a given splitting type µ exist and are given by the values ν * n,p (C µ ). (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 parametric family of (complex-valued) measures ν * n,z on S n for z ∈ P 1 (C) {0, 1}, termed z-splitting measures. The z-splitting measure is a positive probability measure for all integers n ≥ 2 and for all real z = t > n − 1. The uniform distribution on S n is the z-splitting measure for z = ∞ ∈ P 1 (C). (Theorem 2.3). (ii) There exist infinitely many S n -number fields having prescribed splitting types (p i , µ i ) at a given finite set of primes S = {p 1 , ..., p r }, provided that all the splitting types have ν * n,p i (C µ i ) > 0. The latter conditions are satisfied if and only if the S n -number field K has a subring of algebraic integers that is a monogenic order with discriminant relatively prime to i p i . The existence of one such number field K certifies that the associated probability ν * n,p i (µ i ) > 0 (Theorem 2.5). For each n ≥ 2 there is a finite set of exceptional pairs (p i , µ i ) having ν * n,p i (C µ i ) = 0. The exceptional primes p i necessarily satisfy 2 ≤ p i ≤ n − 1, and this set is nonempty for n ≥ 3. The exceptional pairs correspond to the condition that all S n -number fields having such a splitting type (p i , µ) have the prime p i as an essential discriminant divisor (a notion defined in Section 2.3) (Theorem 2.6). The phenomenon of essential discriminant divisors is due to Dedekind [14].

1.2.
Bhargava's number field splitting model. The model for splitting of polynomials studied here parallels a recent probability model of Bhargava [2] for splitting of primes in certain number fields K/Q of degree n, which we discuss in Section 3. 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 and an S n -number field K is a non-Galois extension of Q for n ≥ 3. Bhargava's probability model has sample space the set of all S n -number fields K of discriminant |D K | ≤ D with the uniform distribution, and his results and conjectures concern limiting behavior of splitting densities at a fixed prime p as D → ∞, conditioned on the restriction field K be unramified at (p), i.e. p ∤ D K , the (absolute) field discriminant of K. He formulates conjectures about these limiting distributions for splitting of a fixed prime (p) and proves them for n ≤ 5. These conjectures are open for n ≥ 6.
The connection of S n -number fields to S n -polynomials is that any primitive element θ of an S n -field that is an algebraic integer has θ being a root of an S n -polynomial. Conversely, adjoining 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 polynomial 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 restrictions on the coefficients that the (monic) minimal polynomial of α satisfies.
The limiting densities of the polynomial splitting model for a given prime p do not match the (conjectured) limiting densities of the number field splitting model of Bhargava. The polynomial splitting model distributions ν * n,p depend on p, while the conjectured limiting densities of the Bhargava model are independent of p. Instead Bhargava's limiting densities equal the p → ∞ limit of ν * n,p . In Section 3.2 we present a detailed comparison of the two models.
1.3. Square-free polynomials over F p and homological stability. The splitting measures ν * n,p (C µ ) introduced here count the relative fraction of monic square-free polynomials ( mod p) that have a given factorization type (mod p) in F p [x]. Recently Church, Ellenberg and Farb [6] observed that the monic square-free polynomials (mod p) label points in an interesting moduli space Y n (F p ) defined over F p , the complement of the discriminant locus. (This is a special case of discriminant variety, see Lehrer [30].) They observed that an analogous moduli space Y n (C) can be defined in characteristic 0, defined as a variety over Z, and it has an S n -cover termed configuration space Conf n (C). The configuration space Conf n (C) is an affine variety which is the complement of a set of hyperplanes, which carries an S n -action.
They relate various statistics of the factorizations of square-free polynomials over F p to homological stability properties 1 with respect to S n -actions of the homology of this family of (topological) spaces as both n and p vary. They determine various limiting properties of such statistics as n → ∞ for fixed p, in which the conjugacy class C µ varies with n in specific ways as n increases. In principle their statistics can be expressed in terms of the splitting measure distributions ν * n,p (C µ ). It follows that one can infer certain asymptotic properties of the distributions ν * n,p (C µ ) from their results, and perhaps vice versa. This connection remains to be further explored.
1.4. Contents. Section 2 states the main results. Section 3 discusses Bhargava's number field splitting model and compares its predicted probability distributions with the model of this paper. Section 4 derives basic properties of the splitting probabilities. Section 5 obtains the limiting distributions of splitting probabilities for polynomials with integer coefficients in a box. These splitting probabilities are essentially inherited from the analogous splitting probabilities for random monic polynomials over finite fields. We also establish result (iii) above on existence of infinitely many S n -number fields having given splitting types at a finite set of primes, avoiding exceptional pairs. Section 6 extends the splitting results of this paper to monic polynomials with coefficients in rings of integers of a fixed number 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 ≥ 1. This generalization is an application of results of S. D. Cohen [11]. Section 7 discusses generalizations of the splitting problem to random matrix ensembles.
Notation. Our notation for partitions differs from Macdonald [32]. We denote partitions by µ, where Macdonald uses λ; and the multiplicity of part i of µ is denoted c i (µ) := |{j : µ i = i}|, where Macdonald uses m i (λ). We write a partition µ of n in bracket notation as µ = 1 c 1 , 2 c 2 , · · · , n cn , with only c i > 0 included, following Stanley [40].

Splitting Measures.
The results in this paper are expressible in terms of a discrete family of probability distributions on the symmetric group S n These distributions belong to a continuous family of complex-valued measures on S n . depending on a parameter z ∈ C {0, 1} given below, which we call z-splitting measures. Restricting the parameter to real values z = t ∈ R {0, 1} we obtain signed measures of total mass 1, and all the parameter values t = q = p k which are prime powers give nonnegative probability measures on S n ; these measures originally appeared in statistics 1 For general results on homological stability properties under Sn-actions see Church et al [4], [5], [7].
involving the factorization of random square-free polynomials over F q [X], see Section 4.6.
where µ(d) is the Möbius function.
The necklace polynomial takes integer values at integers, and for positive integers this value has a combinatorial interpretation in terms of necklaces explaining its name, see Section 4.1. These polynomials appear in our context because for X = q = p f a prime power, M m (q) counts the number of irreducible monic degree 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 1 , µ 2 , . . . µ k ) with µ 1 ≥ µ 2 ≥ ... ≥ µ k . Here we regard µ as an unordered partition of n, though for convenience we have listed its elements in decreasing order, and we denote it µ ⊢ n. The conjugacy classes on S n consist of all elements g with a fixed cycle structure and we denote them C µ . For a partition µ ⊢ n we let c i = c i (µ) := |{j : µ j = i}| count its number of parts of size i, and we sometimes denote it by the bracket notation µ = 1 c 1 , 2 c 2 , · · · , n cn , with only c i > 0 included. Definition 2.2. The z-splitting measure ν n,z (g) for g ∈ S n is given by where for a complex number w we interpret w 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, 1. The splitting measure of an individual element g depends only on its cycle type µ = µ(g), so is constant on conjugacy classes C µ of S n . Using the well known formula Properties of these measures are studied in Section 4. The measures are defined by the right side of (2.3) as complex-valued measures for all z on the Riemann sphere, excluding z = 0. The definition implies that they have total mass one, in the sense that g∈Sn ν * n,z (g) = 1.
In this paper we restrict to real values z = t, in which case ν * n,t (g) in general defines a signed measure on S n . In Section 4.4 we prove results specifying positive real z-values where the z-splitting measure is nonnegative. In particular we show nonnegativity of the measure holds for all positive integers t = m ≥ 2.
The z-splitting measures ν * n,z have the following properties, for positive real parameters z = t > 1.
(1) For all real t > n − 1, one has ν * n,t (g) > 0 for all g ∈ S n , For these parameter values ν n,t (·) is a probability measure with full support on S n .
(2) For integers k = 2, 3, ..., n − 1, one has ν * n,k (g) ≥ 0 for all g ∈ S n , so that ν * n,k (·) is a probability measure on S n . For these parameter values this measure does not have full support on S n . It is zero on the conjugacy class of the identity element C 1 n .
In Section 4.5 we prove a complementary result specifying negative real z-values where the z-splitting measure is nonnegative. In particular, nonnegativity holds for all negative integers m ≤ −1 (Theorem 4.5). This result is not used elsewhere in the paper.
We also note that a later result (Theorem 2.6) below refines case (1) of Theorem 2.3 to characterize for each n all pairs (p, µ) with p a prime and ν * n,p (C µ ) = 0. More generally for q = p f , any monic f (x) ∈ F q [x] factors uniquely as where the e i are positive integers and the g i (x) are distinct, monic, irreducible, and non-constant. We may define the splitting type of such a polynomial (following Bhargava [2]) to be the formal symbol where k is the number of distinct irreducible factors of f (x). Here we order the degrees in decreasing order. We let T n denote the set of all possible formal symbols for degree n polynomials, which we call splitting symbols. Thus T 3 = {(111), (21), (3), (1 2 1), (1 3 )}. Using this definition, given any monic f (x) ∈ Z[x] and any prime p, we may assign to it a splitting type µ p (f ) ∈ T n .
In this paper we mainly restrict to square-free splitting types, which are those having all e i = 1. We define T * n ⊂ T n to denote the set of such splitting types. Thus T * 3 = {(111), (21), (3)}. Each element µ := (µ 1 , µ 2 , . . . µ k ) ∈ T * n with µ 1 ≥ µ 2 ≥ ... ≥ µ k specifies a partition of n, to which there is associated a unique conjugacy class C µ ⊂ S n . The conjugacy class C µ is the set of all elements of S n whose cycle lengths are equal to the (unordered) numbers µ 1 , . . . , 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) denote the collection of all degree n monic polynomials with integer coefficients, 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 following properties: (i) The polynomial discriminant (Disc(f ), p) = 1. . (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 constrained by the requirement (1) on the discriminant to belong to T * n , i.e. to be square-free (mod p). For this case we show: Theorem 2.4. (Limiting Splitting Densities) Let n ≥ 2 be given. Then: (1) For each prime p, there holds (2) For each (square-free) splitting type µ ∈ T * n , there holds where ν * n,p is the splitting measure for n with parameter t = p, i.e. .
This result is proved in Section 5; it is a special case r = 1 of Theorem 5.2 which applies more generally to finite sets S = {p 1 , p 2 , ..., p r } of primes. In Section 6 we give a further generalization of the result to algebraic number fields.
2.3. Existence of S n -Number Fields with Prescribed Prime Splitting. We also show there are infinitely many S n -number fields with prescribed prime splitting at any finite set S of primes, of those types allowed by the splitting measures. The splitting measures impose some extra constraints associated to the existence of monogenic orders in the S n -number fields having discriminants relatively prime to given elements.
Theorem 2.5. Let S = {p 1 , ..., p r } denote a finite set of primes, and let U = {µ 1 , ..., µ r } with µ i ⊢ n be a prescribed set of splitting symbols for these primes. Then the following conditions are equivalent.
(1) The positive measure condition (2) There exists an S n -number field K having the following two properties: whose discriminant is relatively prime to p 1 p 2 · · · p r . (P2) The Galois closure K spl of K/Q is unramified at all prime ideals above those in S and the primes in S have prescribed Artin symbols (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 probability measure ν * p,n then has full support on the group S n . However for primes 2 ≤ p < n there are restrictions on the allowed splitting behavior. This restriction has to do with the non-existence of monogenic maximal orders satisfying (P1) for S n -number fields having specific prime factorization at small prime ideals. The polynomials f (x) generating such fields have essential discriminant divisors 2 , as defined in Cohn [12,Defn. 9.55, Lemma 10.44c] and Cohen [8, p. 197]. A famous example due to Dedekind [14] (see [12,Exercise 9.4; Lemma 10.44c]) is an S 3 -number field K for which the prime ideal (2) splits completely in K; all monogenic orders then have an even index, and correspondingly ν * 3,2 ([1 3 ]) = 0. However it is known that infinitely many S 3 -number fields K exist in which the ideal (2) splits completely in the maximal order. This result follows from results of Bhargava for n = 3 discussed in Section 3.1. Such fields are not covered by Theorem 2.5.
Theorem 2.5 allows us to characterize the splitting measures for prime values t = p ≥ 2 having probability 0 in terms of field-theoretic data.
Theorem 2.6. For p a prime, and a splitting type µ ⊢ n, for fixed n ≥ 2, the following three conditions are equivalent.
(C1) The splitting measure at t = p has having a square-free factorization of splitting type C µ . (C3) All S n -number fields K in which (p) is unramified and has splitting type 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 the splitting type µ of (p) in K.
; it does not require f (x) to be an S n -polynomial or to be irreducible over Q. Our proof does not show the existence of even a single field satisfying (C3) for any given pair (p, µ) satisfying (C1). Conjecture 3.2 of Bhargava below would imply that infinitely many such fields exist, and this conjecture is known to be true for n ≤ 5.
In Section 6 we establish generalizations of Theorem 2.5 and Theorem 2.6 above in which the base field 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 are straightforward, using results of S. D. Cohen [11].

Bhargava's number field splitting model
Recall that 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 . Bhargava's number field splitting model has sample space ithe set of all S nnumber fields K of discriminant |D K | ≤ D with the uniform distribution, and his results and conjectures concern the limiting behavior of splitting densities at a fixed prime p as D → ∞, conditioned on the property 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 primes averaged over S n -number fields ordered by the size of their field discriminants. Bhargava developed his conjectures based on the following principle ([2, p. 10]): The expected (weighted) number of global S n -number fields of discriminant D is simply the product of the (weighted) number of local extensions of Q ν that are discriminant-compatible with D, where ν ranges over all places of Q, (finite and infinite).
In this statement a local extension of Q v means a degree nétale algebra E over Q v (not necessarily a field) and discriminant-compatible means that the valuation of the discriminant of E matches that of D and that, in the archimedean case, the signs of the discriminants match. We state two of his conjectures 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 such that: (iii) The degree n field K over Q has normal closure having Galois group S n ; The first conjecture of Bhargava concerns which fraction of S n -number fields have field discriminant K relatively prime to p ([2, Conj. 1.4] ).
where ρ n (p) is the "probability of ramification," given by in which q(k, n) denotes the number of partitions of k into at most n parts.
Bhargava's second conjecture about prime splitting in S n -number fields is as follows ([2, Conj. 1.3]). Conjecture 3.2 (Bhargava). Fix a prime p, a positive integer n, and µ ∈ T * n . Then where ν n (·) denotes the Chebotarev density distribution on conjugacy classes of S n , which is Conjecture 3.2 predicts that the limiting density exists and agrees with that predicted by the Chebotarev density theorem for conjugacy classes (see [28], [36,Chap. 7, §3], [37,Theorem 13.4]); this measure corresponds to the uniform distribution on S n . This limiting distribution depends on n but is independent of p. It is proved for n ≤ 5. The case n = 3 is deducible from results of Davenport and Heilbronn [13], see also Cohen et al [9]. Bhargava proved the result for n = 4 and n = 5 using his earlier results for discriminant density in quartic and quintic fields ( [1], [3]).
For a general viewpoint on Bhargava's conjectures, see Venkatesh and Ellenberg [42, Section 2.3]. Bhargava's conjectures on local mass formulas, were reinterpreted in connection with Galois representations in Kedlaya [25] and further cases were considered by Wood [45], [46].

3.2.
Random polynomial model versus random number field model. We compare the distributions for prime splitting in S n number fields in the random polynomial model against those of the random number field model given in Bhargava's conjectures. These splitting distributions differ.
We summarize the comparison in Table 1. A main feature is that for each n ≥ 1 the densities random monic polynomial model in the p → ∞ limit approaches the uniform density distribution conjectured in Bharagava's model.
Both model predictions assign a weighted contribution of 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 parameter grows. In each model the splitting density is a conditional probability based on conditioning against an "unramifiedness" condition. There is a difference of scale in the cutoffs in the B and D parameters between the two models, in that polynomial discriminants D f grow proportionally to B n . However the limit as the parameters go to infinity, this scale differences play no role.
The main differences in the predicted probabilities in the models are the following.
(1) In Bhargava's conjectures the probability of ramification ρ n (p) depends on both the prime p and the degree n. One has is a quantity θ n (p) which depends on both n and p (Conjecture 3.1) Limiting distribution on S n of splitting types p-splitting distribution ν * n,p (C µ ) on conjugacy classes, whose probabilities depend on both n and p This formula implies that for fixed p and variable n the function ρ n (p) increases to the limit ρ ∞ (p) :

Chebotarev distribution
In contrast, in the random polynomial model the probability of ramification 1 p is independent of n, according to Theorem 2.4 (1). The formula above implies that for fixed n one has ρ n (p) = 1 p + O( 1 p 2 ) as p → ∞, so both ramification probabilities go to 0 as p → ∞ at the same rate.
(2) In Bhargava's conjectures the splitting probabilities are independent of both p and n. In contrast, in the random polynomial model the probabilities ν * n,p (C µ ) depend on both n and p. What features of the models account for the differing answers in the two models? The models themselves have structural differences.
(D1) The (irreducible) polynomial f is associated algebraically not with the ring of integers O F of the field K = Q(θ) generated by a root θ of f , but with the particular monogenic order In particular Disc(f ) may be divisible by primes which do not divide D K , so the "unramified" conditions of the two probability models differ. For some S n -number fields K the ring of integers O K is not monogenic. The number of monogenic orders of a given index in the maximal order O K (isomorphism up to an additive shift of a variable) is known to depend on the index within a given field K, cf. Evertse [16]. (D2) Many different polynomials in F n (B; p) generate the same S n -number field K = K f . Thus each field K that occurs is weighted by the number of polynomials in the box that generate it (and which satisfy the discriminant co-primeness condition). The weights depend in a complicated way on K and B and change as B → ∞.
The difference (D1) of the ramification conditions in the two models presumably accounts for much of the mismatch. The S n -number fields detected by 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 ) with the p-part of D K , with K = Q(θ), which involves studying the p-adic coefficients of f (x). The model of Bhargava is based on a mass formula counting p-adicéetale extensions with weights, and the weights matter. However from the viewpoint of the difference (D2) it is not immediately clear that such weighted sums will conspire to produce the nice limiting values given in Theorem 2.4. To understand difference (D2) better it might be interesting to study an auxiliary question: for each pair of S n -number fields K 1 , K 2 what is the behavior as B → ∞ of the ratio of the number of f (x) in the box of size B that generate the field K 1 (resp. K 2 ) and satisfy p ∤ Disc(f (x)). Does this quantity have a limiting value and if so, how does it depend on K 1 and K 2 ?
To summarize: There are structural differences between the two models. From the viewpoint of each model separately, it is not clear why the other model should have limiting probability densities. It remains to find a more conceptual explanation that accounts for the differences of the limiting densities of the two models.

Splitting Measures
In this section we define and study the one-parameter family of splitting measures ν ⋆ n,z ( z ∈ C) on the symmetric group S n , for each n. We relate this measure at z = q = p k to finite field factorization of degree n monic polynomials over F q . 4.1. Necklace polynomials. The number of monic irreducible polynomials of degree m over finite fields F q for q = p f are well known to be interpolatable by universal polynomial M m (X) evaluated at value X = q. Recall that for m ≥ 1 the necklace polynomial of degree m is M m (X) ∈ Q[X] by where µ(d) is the Möbius function. For m = 0 we set M 0 (X) = 1. We note that M 1 (X) = X and M 2 (X) = 1 2 X(X − 1). The value M m (k) for positive integer k has a combinatorial interpretation as counting the number of necklaces of m distinct colored beads formed using k colors which have the property of being primitive in the sense that their cyclic rotations are distinct (Moreau [35]). The name "necklace polynomial" was proposed by Metropolis and Rota [34]. Clearly M m (X) ∈ 1 m Z[X], for m ≥ 1. These polynomials are discussed in Hazewinkel [23,Sect. 17] in connection with Witt vectors.
We give some basic properties of M m (X).   [33]. A proof is given in Rosen [38, p. 13].
(2) To verify that a polynomial in 1 m Z[X] is integer-valued, it suffices to check the integrality property holds at m consecutive integer values of X. The integrality property at positive integers follows from the counting interpretation of the values M m (j) of Moreau [35], see [34].
We will use the following bounds on the size of M m (X). One has (X − 1) 2 ∤ M m (X) for all m ≥ 2.
In addition, for 1 ≤ j ≤ m there holds for real t > m − 1, In addition, for each m ≥ 2 and t > 0 with t(t + 1) > m − 2, there holds for Proof.
(1) We have M m (0) = 0 since it has no constant term for m ≥ 1. For (2) For m ≥ 2 and real t ≥ 2, one has For the second part, suppose m ≥ 2 and 1 ≤ j ≤ m. We have for j = 1 and t > m − 1 that M 1 (t) = t > m − 1. For j = 2 and t > m − 1 we have the last inequality being immediate for m = 2 and easy for m ≥ 3. Finally, for 3 ≤ j ≤ m, and t > m − 1, we have t j − t j 2 +1 ≥ 1, whence by (4.5), which gives (4.4) in this case.
For the second part, it suffices to show for 1 ≤ j ≤ m/2 that For 2j = 2 we have by hypothesis For 2j ≥ 4 and m ≥ 6, the condition t(t + 1) > m − 2 implies t > 2. Then (4.6) applies and we obtain.

Cycle polynomials.
To any partition µ ⊢ n we associate the cycle polynomial Here N µ (X) ∈ Q[X] is a polynomial of degree n (since n j=1 jc j = n). The values N µ (X) for prime powers X = q = p f count the number of square-free polynomial factorizations of type µ in F q [X], as shown in Section 4.6. ( (3) The polynomial N µ (X) is divisible by X m , where m ≥ 1 counts the number of distinct cycle lengths appearing in µ.
(2) The property follows by direct calculation of the top degree term in (4.7).
(4) Both sides of the identity (4.8) are polynomials of degree n, so it suffices to verify that the identity holds at n + 1 distinct points X = n j in C. To this end, we make use of a combinatorial interpretation of N µ (X) for X = q = p k a prime power, given in Proposition 4.6 below. The sum on the left evaluated at 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 discriminant over F q . The resulting polynomial F (X) satisfies F (q) = q n − q n−1 , according to Proposition 4.6 (1), verifying the identity at X = q.

Splitting Measures.
For each n ≥ 2 we define the parametric family of (necklace) splitting measures ν * n,z on the symmetric group S n , with family parameter z ∈ C, by means of their values on conjugacy classes . (4.9) For any element g ∈ C µ we set The latter formula coincides with the definition (2.1) for ν * n,z (g). Since this formula 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 conjugacy classes, for all z on the Riemann sphere C := C ∪ {∞} except possibly at z = 0, 1. The measure at z = ∞ is the uniform measure, ν n,∞ (g) = 1 n! , a result 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 a signed measure. It is studied by the first author in [27].
We next show that these measures have total (complex-valued) mass one.
Proof. For z ∈ C {0, 1, ∞} the lemma follows from the normalization identity (4.8) for the cycle polynomials. It extends by analytic continuation in z to the values z = 1, ∞.

Nonnegativity conditions for splitting measures: Positive real
z. This paper is (only) with the case that z = t is a real number (t = 0, 1), In this case the measure is real-valued, and is a signed measure, of total (signed) mass one by Proposition 4.4. We now treat ranges of positive real z and prove Theorem 2.3, which specifies several real parameter ranges where these measures are nonnegative, and so define probability measures; these parameter values include all integer values z = m ≥ 2.
Proof of Theorem 2.3. To decide on nonnegativity or positivity of ν * n,t (C µ ), it suffices to study the individual terms M j (t) c j (µ) for 1 ≤ j ≤ n and to show nonnegativity (resp. positivity) of each of the numerators (4.11) (1) We verify that for t > n − 1 and all µ, all terms in the definition (4.9) of ν * n,t (C µ ) for 1 ≤ j ≤ n have M j (t) c j (µ) > 0. The positivity of the terms in (4.11) is immediate for n = 1 so suppose n ≥ 2. Using Lemma 4.2 (2) for t > n − 1 we have whence all factors in the product (4.11) are positive, as asserted.
(2) For each integer 2 ≤ k ≤ n − 1, the normalizing factor 1 k n−1 (k−1) in the definition is positive. Since M j (X) is an integer-valued polynomial for all j ≥ 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 1 n = {e}, for 2 ≤ k ≤ n − 1 we have that ν * n,k (C 1 n ) = 0, since in these case the j = 1 factor in (4.11) has (M 1 (k)) n = 0.
(3) The limit as t → ∞ is driven by the lead term asymptotics of the polynomial M j (t). Using j jc j (µ) = n and M j (t) = 1 j t j + O(t j−1 ) we obtain as asserted.

4.5.
Nonnegativity conditions for splitting measures: negative real z. We prove complementary results specifying some negative real parameter values z = −t < 0 where µ n,−t (C µ ) is nonnegative and so defines a probability measure.
Theorem 4.5. Let n ≥ 2. Then for real z = −t < 0 the signed measures ν * n,t on S n have the following properties.
(1) For all real values t > 0 having t(t + 1) > n − 2, the measure ν * n,−t on S n is strictly positive, so that it defines a probability measure on S n with full support.
(2) For all integers k ≥ 1 having t(t + 1) ≤ n − 2 the measure ν n,−k is nonnegative and defines a probability measure on S n . This measure does not have full support. it is zero on the conjugacy class C µ with µ = 2 n/2 if n is even, and on the conjugacy class with µ = 1, 2 (n−1)/2 if n is odd.

Proof.
(1) To show positivity of the measure we keep track of the signs of all the factors in the definition (4.9). Since t > 0 the prefactor has sign

Lemma 4.2 (3) then gives for
Since m j (µ) ≤ ⌊ n j ⌋, we obtain the positivity of all even degree terms.
It remains to show that all M 2j+1 (−t) < 0. This holds for t ≥ 2 by Lemma 4.2 (3), and t ≥ 2 holds for n ≥ 8. For the remaining cases we check M 1 (−t) = −t < 0 for t > 0, and that for 2j+1 = 3, 5, 7 we have M 2j+1 (−t) = −t 2j+1 + t < 0 for t > 1, (2) To show nonnegativity of the measure ν * n,−k for those positive integer k with k(k − 1) ≤ n − 2, the argument of (1) s applies with the following changes. For even indices 2j, we use the fact that M 2j (−k) is a positive integer, so either the descending factorial remains positive or else is zero if a zero is encountered. So the sign of this term may be treated as positive. For the odd indices 2j + 1, either the initial value M 2j+1 (−k) = 0, in which case the measure is 0, or else M 2j+1 (−k) < 0 and the sign argument above applies. One has M 2j+1 (−k) < 0 if k ≤ 2 so the only problematic value is M 2j+1 (−1). This value is always 0, as may be checked. Thus nonnegativity of the measure follows.
It remains to show that the measure does not have full support. We verify for n = 2ℓ that for µ = 2 ℓ , one has ν * n,−k (C µ ) = 0 for all positive integers k with k(k + 1) ≤ n − 2. Here m 2 (µ) = ℓ and the integer so that the descending factorial (M 2 (−m)) ℓ = 0. One verifies similarly that for n = 2ℓ + 1 ≥ 3 one has ν n,−k (C µ ) = 0 for µ = 1, 2 ℓ , where again m 2 (µ) = ℓ and n−2 2 = ℓ − 1. 4.6. Splitting measures and finite field factorizations. We now give a rationale for study of z-splitting measures. When z = q = p k is a prime power, these measures occur in the statistics of factorization of monic polynomials of degree n in F q [X], drawn from a uniforml distribution, conditioned on being square-free.
Recall that a monic polynomial where the e i are positive integers and the g i (x) are distinct, monic, irreducible, and non-constant. We have the following basic facts about square-free factorizations.
Proposition 4.6. Fix a prime p ≥ 2, 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 .
(1) Exactly q n−1 of these polynomials have discriminant Disc(f ) = 0 in F q . Equivalently, exactly q n−1 of these polynomials are not square-free when factored into irreducible factors over F q [X].
(2) The number N (µ; q) of f (x) ∈ F n,q whose factorization over F q into irreducible factors is square-free of degree type µ := (µ 1 , ..., µ r ), with µ 1 ≥ µ 2 · · · ≥ µ r with µ ⊢ n, having c j = c j (µ) factors of degree j satisfies (4.12) in which N µ (X) denotes the cycle polynomial for µ. This proposition has the following consequence, giving a realization of the z-splitting measure for z = q = p k a prime power.
Proposition 4.7. Consider a random monic polynomial g(X) of degree n drawn from F q [x] with the uniform distribution, where q = p f . Then the probability of g(x) having a factorization into irreducible factors of splitting type µ ∈ T * n , conditioned on g(x) having a square-free factorization, is exactly 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 4.6 (1), (2) together evaluate the conditional probability Comparing the right side with the definition (2.3) of the splitting measure shows that it equals ν * n,q (C µ ).

5.1.
Counting monic S n -polynomials with coefficients in a box. It is 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 was proved in 1936 by van der Waerden [43], who showed that the fraction of all monic 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 → ∞. An improved quantitative form of this assertion was given in 1973 by Gallagher [20], which we formulate as follows. Indeed, we have Disc(f ) = 1≤i<j≤n (θ i − θ j ) 2 , so it suffices to show that f (x) ∈ F n (B) have all roots |θ i | < 2B. This holds because if some root |θ| ≥ 2B then |θ n−j | ≤ |θ| n /(2B) j , whence which contradicts θ being a root of f (x). The error estimate in Gallagher's estimate was recently improved by Dietmann [15] to Improvements of Gallagher's results in some other directions are given in Zywina [47].

5.2.
Density of S n -polynomials with specified splitting types. Our object is to refine the result above by counting the number of such polynomials generating an S n -extension that have a given splitting type at a finite set of primes. As above, for integer B let F n (B) denote the set of monic polynomials of degree n with coefficients −B < a i ≤ B, so that #{f (x) ∈ F n (B)} = (2B) n . Theorem 2.4 is the special case r = 1 of the following result.
We note that the condition gcd(Disc(f ), r i=1 p i ) = 1 on a monic irreducible polynomial guarantees that the field K = Q(θ) generated by a single root of f (x) is unramified over all the primes in S. In that case, the discriminant Disc(f ) detects the discriminant of the ring O f = Z[1, θ, ..., θ n−1 ] , 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 have 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 ), r i=1 p i ) = 1.
Proof. For each prime p the behavior of Disc(f ) (mod p) is determined by (a 0 , a 1 , ..., a n−1 ) (mod p). Thus if M divides B then Proposition 4.6 (1) shows that exactly a fraction of 1 p of these polynomials have Disc(f ) ≡ 0 (mod p). The polynomials are labelled by lattice points in the closed box [−B + 1, B] n , and we call a lattice point admissible if it corresponds to a polynomial in F n (B, S). For a general B we first round down to a box of side B ′ = M ⌊ B M ⌋, and using there the Chinese remainder theorem we find exactly (2B ′ ) n r i=1 (1 − 1 p i ) admissible polynomials in the smaller box belong to F n (B, S). This number undercounts (2B) n r Similarly we may round up to a box of side B ′′ = M ⌈ B M ⌉ and using there a similar argument we find exactly We now use the inequality, valid for real |x| ≤ 1 2n , ).
(5.6) This yields the estimate Proof. Let For splitting types on box of side B ′ = M ⌊ B M ⌋ by reduction (mod M ) together with Proposition 4.6 (2) and the Chinese remainder theorem we get a product distribution of all splitting types (mod p i ) for 1 ≤ i ≤ r, where N µ i (·) is a cycle polynomial. We have a similar formula for an enclosing box of side B ′′ = M ⌈ B M ⌉, with (2B ′′ ) n replacing (2B ′ ) n . Assuming B ≥ 2nM we obtain by an application of (5.6) that Next we note that 1 q n N µ (q) = 1 − 1 q )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 For B ≥ 2nM , we replace (2B) n i (1 − 1 p i ) with |F n (B; S)| using (5.7) we obtain The result follows on dividing both sides by Remark. The conclusion in Theorem 5.2 is insensitive to the shape of the box bounding the coefficients as long as the box increases homothetically as B → ∞, e.g. one can use −c n,j B < a j < c n,j B, where c n,j are positive constants independent of B, and derive exactly the same limiting formula. For example, c n,j = n j 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 symmetric group S n as a Galois group, represented as a permutation group acting transitively on the roots of a polynomial, that distinguishes it from some of its subgroups. Let G be a permutation group G ⊂ S n (i.e. a permutation representation of the abstract group G). The elements in a conjugacy class in G necessarily have the same cycle type as permutations, but the converse need not hold. That is, the cycle type of a conjugacy class in G need not determine it uniquely. For example, the group G = {(12)(34), (13)(24), (14)(23), (1)(2)(3)(4)} ⊂ S 4 is abelian so all conjugacy classes have size 1 but three of these classes have identical cycle structures. This uniqueness property does hold for cycle types for the full symmetric 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 of it.
Proof of Theorem 2.5. (1) ⇒ (3). By hypothesis we are given S = {p 1 , ..., p r } and splitting types U = {µ 1 , .., µ r } with the property that all ν * n,p i (C µ i ) > 0. We will show the number of S n -number fields K whose Galois closure K ′ has the given Artin symbols is infinite, by showing it is arbitrarily large. Since the splitting type of a polynomial f (X) generating an S n -number field modulo p determines the corresponding Artin symbol, it suffices to specify factorizations of polynomials (mod p i ) which we can do using Theorem 5.2. Given k ≥ 1 we choose S k := S S * k with S * k = {p r+1 , · · · , p r+k } being a 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 splitting symbols U k := {µ r+1 , ..., µ r+k } arbitrarily in S n for the auxiliary primes and the condition ν * n,p r+j (C µ r+j ) > 0 will automatically hold by Theorem 2.3 (2). The square-free condition on the polynomial modulo each p i guarantees that the polynomial discriminant is relatively prime to p 1 p 2 · · · p r+k and this property guarantees that (P1) holds. Theorem 5.2 now implies the existence of 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 at least one such S n -number field K exhibiting the given splitting behavior. Since each S n for n ≥ 2 has at least two distinct conjugacy classes, we obtain in this way at least 2 k different S n -number fields, all of which match the splitting types C µ i for 1 ≤ i ≤ r in (5.8) and which are distinguishable among themselves by how the auxiliary primes p r+j , 1 ≤ j ≤ k split. Since k can be arbitrarily large, the result follows.
(2) ⇒ (1). By hypothesis the given field K possesses is a monogenic order Z[1, θ, ..., θ n−1 ] satisfying (P1). The minimal polynomial for θ is then a monic polynomial f (x) ∈ Z[X] which satisfies gcd(Disc(f ), p 1 · · · p r ) = 1. This polynomial then has square-free factorization (modp i ) yielding the splitting types C µ i for 1 ≤ i ≤ r, see 4 Lang [29, I. §8, Proposition 25]. We next observe that the splitting type conditions are congruence conditions ( mod p 1 · · · p n ) on the coefficients of f , and they enforce the condition gcd(Disc(f ), p 1 p 2 · · · p r ) = 1. These congruence conditions are satisfied for a positive proportion of polynomials in the box, so in Theorem 5.2 the left side of (5.5) is positive, which certifies that each ν * n,p i (C µ i ) > 0.
Proof of Theorem 2.6. (C1) ⇔ (C2). Since p = 0, 1 the condition ν * n,p (C µ ) = 0 holds if and only if N µ (p) = 0. By Proposition 4.6 (2) the latter condition 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. The latter condition is exactly (C2).
(C1) ⇔ (C3). We establish the contrapositive. Suppose (C3) does not hold. This says that there exists a S n -number field K which at (p) is unramified and has splitting type µ. Now the equivalence of Theorem 2.6 applied for a single prime p 1 = p shows that ν * n,p (µ) > 0, which is equivalent to the condition that (C1) does not hold.
We remark that this argument does not establish whether or not there exist any S n extensions K which satisfy condition (C3) for given splitting data µ.

Number of S n -Polynomials with Specified Splitting Types over Number Fields
We consider polynomials with coefficients drawn from an algebraic number field k, not necessarily Galois over Q. We set [k : Q] = d, and say that an extension L/k with [L : k] = n is a relative S n -number field if the Galois closure L ′ of L over k has Gal(L ′ /k) ≃ S n . We let D k denote the absolute discriminant of k over Q.
Let O k denote the ring of algebraic integers in k. We consider monic polynomials 4 The hypothesis of Lang's Proposition 25 requires Z[1, θ, · · · , θ n−1 ] to be integrally closed, i.e. the full ring of integers O k . As he notes, the argument can be done by localizing over each prime ideal (pi), and here (Disc(f ), pi) = 1 implies that the integral closure condition holds locally. with all α j ∈ O k . Choose an integral basis O k = Z[ω 1 , ω 2 , ..., ω d ], and let Ω = (ω 1 , ..., ω d ) denote this (ordered) integral basis. We now have 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 (2B) nd polynomials in the box.
Next we let S = {p 1 , ..., p r } denote a finite ordered set of (distinct) prime ideals in O k . We allow different ideals in the list to have residue class fields of 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 1 , ..., µ r } denote a finite ordered set of splitting types of S n (the different µ j need not be distinct).
Theorem 6.1. Suppose that k/Q is a number field, not necessarily Galois over Q, Let S = {p 1 , ..., p r } be an ordered finite set of distinct prime ideals in 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 1 , ..., µ r } is a given ordered set of splitting symbols. Then for fixed n ≥ 2, the following hold.
(1) Let F n (B; S, Ω) denote the set of all degree n polynomials f (x) in (ii) f (x) has relative splitting field K f over k that is an S n -extension of k. Then Proof. This result parallels the proof of Theorem 5.2. We only sketch the details, indicating the main changes needed. Suppose [k : Q] = d. Firstly, we have #{f (x) ∈ F n (B; Ω)} = (2B) nd .
The condition for the polynomial discriminant gcd(Disc(f ), r i=1 p i ) = (1) is exactly that the polynomial f (x) have square-free factorization ( mod p i ) For the limit in (1) we obtain an exact count when going through boxes having all sides B = M m for some integer m ≥ 1, which is For each prime ideal p i this holds using Proposition 4.6 (1) since we have an integral multiple of complete residue systems (mod p i ) in the box, and it holds for all p i simultaneously using the Chinese remainder theorem for the box. Allowing a general B adjusts this formula by a multiplicative amount 1 + O( ndM B ), 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 is This result follows using an upper bound of Cohen [11,Theorem 2.1], in his result specifying that F t (x) = X n + n−1 i=0 t i X i , that K = k, and noting the Galois group G = S n for F t (X) over the function field k(t 1 , · · · , t n ).
Thirdly, on restricting the box size to the special form B = M m with m ≥ 1, one gets an exact count This formula is equivalent to Changing the box size to an arbitrary integer B introduces at most a multiplicative roundoff error of 1 + O( ndM B ). Fourthly, we combine the above estimates to obtain an analogue of Theorem 5.4, stating that The formula (6.2) follows on letting B → ∞.
Remark. The conclusion in Theorem 6.1 is insensitive to the shape of the box bounding 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 relative S n -number fields K over k having prescribed splitting above a given finite 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 convention that a relative S n -number field K over k is a degree n extension of k whose Galois closure over k has Galois group S n . We recall that the (relative) discriminant .., p r } denote a finite set of prime ideals of k. and let U = {µ 1 , ..., µ r } with µ j ⊢ n be a prescribed set of splitting symbols for these prime ideals. Set q i = N k/Q p i . Then the following conditions are equivalent.
(2) There exists a relative S n -number field K/k having the following two properties: (P1-k) The field K contains a monogenic order O = O k [1, θ, ...θ n−1 ] whose relative discriminant Disc(O|O k ) is relatively prime to p 1 p 2 · · · p r . (P2-k) The Galois closure K spl of K over k is unramified at all prime ideals above those in S and the primes in S have prescribed Artin symbols (3) There exist infinitely many relative S n -number fields K over k having properties (P1-k) and (P2-k).
Proof. The proof parallels that of Theorem 2.5, using Theorem 6.1 in place of 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 polynomial discriminants and relative discriminants of the associated monogenic order in K = k(θ), for θ a root of f (x). That is, setting O f := O k [1, θ, · · · , θ n−1 ]. one has the equality of O k -ideals; here (Disc(f )) is a principal ideal. We use this fact to show that (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 , and positive density follows by Theorem 6.1 since all conditions imposed are congruence conditions.
To conclude the paper we formulate a generalization of Theorem 2.6. For a 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 relative discriminants Disc(O|O k ) all monogenic orders O := O k ]1, θ, · · · , θ n−1 ] of the 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 ≥ 2, 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 coefficients in O k whose (mod p) factorization has splitting type µ, the relative discriminant Disc(O f |O k ) is divisible by p. (C3-k) All relative S n -extensions K of k in which p is unramified and has splitting type µ necessarily have p as an essential relative discriminant divisor.
Proof. The proof parallels that of Theorem 2.6. We note only that to establish the equivalence (C1-k) ⇔ (C2-k), one uses (6.3).
In cases where (C1-k) holds this proof does not establish that there exist any fields satisfying (C3-k).

Generalizations
The problem studied in this paper can be viewed as a special case of study of characteristic polynomials of random matrices. One may consider random matrices drawn from a group like GL(n, Z) with constraints on the size of the matrix A = [a i,j ] ( measured in some matrix norm), and also putting side conditions on the allowed elements. The problem for degree n polynomials 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 having characteristic polynomial f (x). After reduction (mod p) from GL(n, Z) one obtains a particular distribution of random matrices having entries over the finite field F p with a side condition forcing many matrix entries to be zero. Our imposed restriction on factorization of polynomials being squarefree corresponds requiring that the associated matrices in GL(n; F p ) have distinct eigenvalues, i.e. they belong to semisimple conjugacy classes. One can ask whether there are further interesting generalizations of the model of this paper results in the random matrix context.
There are many results known using random integer matrices in general models. In 2008 Kowalski [26,Chap. 7] showed that the characteristic polynomial of a random matrix in SL(n, Z) drawn using a random walk is an S n -polynomial with probability approaching 1 as the number of steps increases. For splitting fields of characteristic polynomials of random elements drawn from more general split reductive arithmetic groups G see work of Gorodnik and Nevo [22], Jouve, Kowalski and Zywina [24]. In their 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 [31] give a further generalization to a wider class of groups with coefficients in a wider class of fields, where the "generic" Galois group of a random element may have a more complicated behavior.
There are also many results known on the distribution of characteristic polynomials of random matrices over finite fields F q ; this subject is surveyed in Fulman [18]. His paper puts emphasis on M at(n, F q ) and GL(n, F q ), and includes 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 depending on 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 probability measures M GL,u,q on conjugacy classes of GL(n, F q ), which when conditioned on fixed n do not depend on the parameter u and have the rational function interpolation property in the parameter q. They therefore extend to a complex parameter z, defining complex-valued measures M GL,z,q . He remarks [18,Section 3.3] that this distribution coincides with the distribution on partitions describing the Jordan block structure of a random unipotent element of GL(n, q). It would be interesting to determine whether the measures M GL,u,q have any relation to the splitting measures studied in this paper.