For various nonsolvable groups G, we prove the existence of extensions of the rationals Q with Galois group G and inertia groups of order dividing ge(G), where ge(G) is the smallest exponent of a generating set for G. For these groups G, this gives the existence of number fields of degree ge(G) with an unramified G-extension. The existence of such extensions over Q for all finite groups would imply that, for every finite group G, there exists a quadratic number field admitting an unramified G-extension, as was recently conjectured. We also provide further evidence for the existence of such extensions for all finite groups, by proving their existence when Q is replaced with a function field k(t) where k is an ample field.

There are several variants of the inverse Galois problem which involve restrictions on ramification. In this paper, we give sufficient conditions that a given finite group [Formula: see text] occurs infinitely often as a Galois group over the rationals [Formula: see text] with all nontrivial inertia groups of order [Formula: see text]. Notably any such realization of [Formula: see text] can be translated up to a quadratic field over which the corresponding realization of [Formula: see text] is unramified. The sufficient conditions are imposed on a parametric polynomial with Galois group [Formula: see text] — if such a polynomial is available — and the infinitely many realizations come from infinitely many specializations of the parameter in the polynomial. This will be applied to the three finite simple groups [Formula: see text], [Formula: see text] and [Formula: see text]. Finally, the applications to [Formula: see text] and [Formula: see text] are used to prove the existence of infinitely many optimally intersective realizations of these groups over the rational numbers (proved for [Formula: see text] by the first author in [J. König, On intersective polynomials with nonsolvable Galois group, Comm. Alg. 46(6) (2018) 2405–2416.

The existence of finite dimensional central division algebras with no maximal subfield that is Galois over the center (called noncrossed products), was for a time the biggest open problem in the theory of division algebras, before it was settled by Amitsur. Motivated by Brussel’s discovery of noncrossed products over Q((t)), we describe the “location” of noncrossed products in the Brauer group of general Henselian valued fields with arbitrary value group and global residue field. We show that within the fibers defined canonically by Witt’s decomposition of the Brauer group of such fields, crossed products and noncrossed products are, roughly speaking, separated by an index bound. This generalizes a result of the first and third author for rank 1 valued Henselian fields. Furthermore, we prove that all fibers which are not covered by the rank 1 case, and where the characteristic of the residue field does not interfere, contain noncrossed products. We show by example that, unlike in the rank 1 case, the value of the index bound does not depend on the number of roots of unity that are present. Thus, the index bounds are in general of a different nature than in the rank 1 case.

It has been conjectured by Sarkozy that with finitely many exceptions, the set of quadratic residues modulo a prime $p$ cannot be represented as a sumset $\{a+b\colon a\in A, b\in B\}$ with non-singleton sets $A,B\subset F_p$. The case $A=B$ of this conjecture has been recently established by Shkredov. The analogous problem for differences remains open: is it true that for all sufficiently large primes $p$, the set of quadratic residues modulo $p$ is not of the form $\{a'-a\colon a',a\in A,\,a'\ne a\}$ with $A\subset F_p$? We attack here a presumably more tractable variant of this problem, which is to show that there is no $A\subset F_p$ such that every quadratic residue has a \emph{unique}representation as $a'-a$ with $a',a\in A$, and no non-residue is represented in this form. We give a number of necessary conditions for the existence of such $A$, involving for the most part the behavior of primes dividing $p-1$. These conditions enable us to rule out all primes $p$ in the range $13

This paper proves the existence of infinitely many distinct Galois realizations of the alternating group $A_5$ over $dQ$ which are optimally intersective, i.e. Galois realizations each of which is the splitting field of a polynomial which has a root in $dQ_p$ for all primes $p$ and is the product of exactly two nonlinear factors. This result is known for $A_4$.

In this paper, an intersective polynomial is a monic polynomial in one variable with rational integer coefficients, with no rational root and having a root modulo m for all positive integers m. Let G be a finite noncyclic group and let r(G) be the smallest number of irreducible factors of an intersective polynomial with Galois group G over \(\mathbb {Q}\). Let s(G) be smallest number of proper subgroups of G having the property that the union of their conjugates is G and the intersection of all their conjugates is trivial. It is known that \(s(G)\le r(G)\). It is also known that if G is realizable as a Galois group over the rationals, then it is also realizable as the Galois group of an intersective polynomial. However it is not known, in general, whether there exists such a polynomial which is a product of the smallest feasible number s(G) of irreducible factors. In this paper, we study the case \(G=S_n\), the symmetric group on n letters. We prove that for every n, either \(r(S_n)=s(S_n)\) or \(r(S_n)=s(S_n)+1\) and that the optimal value \(s(S_n)\) is indeed attained for all odd n and for some even n. Moreover, we compute \(r(S_n)\) when n is the product of at most two odd primes and we give general upper and lower bounds for \(r(S_n)\).

There has been interest during the last decade in properties of the sequence gcd(a − 1, b − 1), n = 1, 2, 3, ..., where a, b are fixed (multiplicatively independent) elements in one of Z,C[T ], or Fq[T ]. In the case of Z, Bugeaud, Corvaja and Zannier have obtained an upper bound exp( n) for any given > 0 and all large n, and demonstrate its sharpness by extracting from a paper of Adleman, Pomerance, and Rumely a lower bound exp(exp(c log n log log n )) for infinitely many n, where c is an absolute constant. Silverman has proved an analogous lower bound deg gcd(a−1, b−1) ≥ cn for infinitely many n, over Fq[T ]. This paper generalizes Silverman’s theorem to gcd(Φm(a),Φm(b)) for any positive integer m, where Φm(x) is the mth cyclotomic polynomial, Silverman’s result being the case m = 1. Over Z, the lower bound has been proved in the first author’s Ph.D. thesis for the case m = 2, i.e. for gcd(a + 1, b + 1). Here we prove a 2000 Mathematics Subject Classification. Primary 11A05; Secondary 11R47,11N37. Mots clefs. greatest common divisor, sequence, cyclotomic polynomial. 2 Joseph Cohen, Jack Sonn conditional result that the lower bound for arbitrary m holds over Z under GRH (the generalized Riemann Hypothesis).

The existence of finite dimensional central division algebras with no maximal subfield that is Galois over the center (called noncrossed products), was for a time the biggest open problem in the theory of division algebras, before it was settled by Amitsur. Motivated by Brussel's discovery of noncrossed products over Q((t)), we describe the "location" of noncrossed products in the Brauer group of general Henselian valued fields with arbitrary value group and global residue field. We show that within the fibers defined canonically by Witt's decomposition of the Brauer group of such fields, crossed products and noncrossed products are, roughly speaking, separated by an index bound. This generalizes a result of the first and third author for rank 1 valued Henselian fields. Furthermore, we prove that all fibers which are not covered by the rank 1 case, and where the characteristic of the residue field does not interfere, contain noncrossed products. We show by example that, unlike in the rank 1 case, the value of the index bound does not depend on the number of roots of unity that are present. Thus, the index bounds are in general of a different nature than in the rank 1 case.

Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q_p for all p, or equivalently, with roots mod n for all n. It is known that f(x) cannot be irreducible but can be a product of two or more irreducible polynomials, and that if f(x) is a product of m>1 irreducible polynomials, then its Galois group must be m-coverable, i.e. a union of conjugates of m proper subgroups, whose total intersection is trivial. We are thus led to a variant of the inverse Galois problem: given an m-coverable finite group G, find a Galois realization of G over the rationals Q by a polynomial f(x) in Z[x] which is a product of m nonlinear irreducible factors (in Q[x]) such that f(x) has a root in Q_p for all p. The minimal value m=2 is of special interest. It is known that the symmetric group S_n is 2-coverable if and only if 2

There has been interest during the last decade in properties of the sequence gcd(a^n-1,b^n-1), n=1,2,3,..., where a,b are fixed (multiplicatively independent) elements in either the rational integers, the polynomials in one variable over the complex numbers, or the polynomials in one variable over a finite field. In the case of the rational integers, Bugeaud, Corvaja and Zannier have obtained an upper bound exp(ϵn) for any given ϵ>0 and all large n, and demonstrate its approximate sharpness by extracting from a paper of Adleman, Pomerance, and Rumely a lower bound exp(exp(clog n/loglog n)) for infinitely many n, where c is an absolute constant. The upper bound generalizes immediately to gcd(Φ_N(a^n), Φ_N(b^n)) for any positive integer N, where Φ_N(x)is the Nth cyclotomic polynomial, the preceding being the case N=1. The lower bound has been generalized in the first author's Ph.D. thesis to N=2. In this paper we generalize the lower bound for arbitrary N but under GRH (the generalized Riemann Hypothesis). The analogue of the lower bound result for gcd(a^n-1,b^n-1) over F_q[T] was proved by Silverman; we prove a corresponding generalization (without GRH).

Since Amitsur’s discovery of noncrossed product division algebras in 1972, their existence over more familiar fields has been an object of investigation. Brussel’s work was a culmination of this effort, exhibiting noncrossed products over the rational function field k ( t ) and the Laurent series field k (( t )) over any global field k —the smallest possible centers of noncrossed products. Witt’s theorem gives a transparent description of the Brauer group of k (( t )) as the direct sum of the Brauer group of k and the character group of the absolute Galois group of k . We classify the Brauer classes over k (( t )) containing noncrossed products by analyzing the fiber over χ for each character χ in Witt’s theorem. In this way, a picture of the partition of the Brauer group into crossed products/noncrossed products is obtained, which is in principle ruled by a relation between index and number of roots of unity. As a side consequence of the result there are crossed products that have a noncrossed product primary component.

It is now known that for any prime p and any finite semiabelian p-group G, there exists a (tame) realization of G as a Galois group over the rationals Q with exactly d = (d(G) ramified primes, where d(G) is the minimal number of generators of G, which solves the minimal ramification problem for finite semiabelian p-groups. We generalize this result to obtain a theorem on finite semiabelian groups and derive the solution to the minimal ramification problem for a certain family of semiabelian groups that includes all finite nilpotent semiabelian groups G. Finally, we give some indication of the depth of the minimal ramification problem for semiabelian groups not covered by our theorem.

A (monic) polynomial f(x)∈ℤ[x] is called intersective if the congruence f(x)≡0 mod m has a solution for all positive integers m. Call f(x) nontrivially intersective if it is intersective and has no rational root. It was proved by the author that every finite noncyclic solvable group G can be realized as the Galois group over ℚ of a nontrivially intersective polynomial (noncyclic is a necessary condition). Our first remark is the observation that the corresponding result for nonsolvable G reduces to the ordinary inverse Galois problem for G over ℚ. The second remark has to do with the scarcity of explicit examples of nontrivial intersective polynomials with given Galois group, and gives the first known example for the dihedral group of order ten.

Since Amitsur's discovery of noncrossed product division algebras more than 35 years ago, their existence over more familiar fields has been an object of investigation. Brussel's work was a culmination of this effort, exhibiting noncrossed products over the rational function field k(t) and the Laurent series field k((t)) over any global field k -- the smallest possible centers of noncrossed products. Witt's theorem gives a transparent description of the Brauer group of k((t)) as the direct sum of the Brauer group of k and the character group of the absolute Galois group of k. We classify the Brauer classes over k((t)) containing noncrossed products by analyzing the fiber over chi for each character chi in Witt's theorem. In this way, a picture of the partition of the Brauer group into crossed products/noncrossed products is obtained, which is in principle ruled solely by a relation between index and number of roots of unity. For large indices the noncrossed products occur with a "natural density" equal to 1.

Since Amitsur’s discovery of noncrossed product division algebras more than 35 years ago, their existence over more familiar fields has been an object of investigation. Brussel’s work was a culmination of this effort, exhibiting noncrossed products over the rational function field k(t) and the Laurent series field k((t)) over any global field k — the smallest possible centers of noncrossed products. Witt’s theorem gives a transparent description of the Brauer group of k((t)) as the direct sum of the Brauer group of k and the character group of the absolute Galois group of k. We classify the Brauer classes over k((t)) containing noncrossed products by analyzing the fiber over χ for each character χ in Witt’s theorem. In this way, a picture of the partition of the Brauer group into crossed products/noncrossed products is obtained, which is in principle ruled solely by a relation between index and number of roots of unity. For large indices the noncrossed products occur with a “natural density” equal to 1.

Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q(p) for all p, or equivalently, with roots mod n for all n. It is known that f(x) cannot be irreducible but can be a product of two or more irreducible polynomials, and that if f(x) is a product of m > 1 irreducible polynomials, then its Galois group must be a union of conjugates of m proper subgroups. We prove that for any m > 1, every finite solvable group that is a union of conjugates of m proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with m = 2) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric, i.e. regular, extension of Q(t).

It is now known [H. Kisilevsky, J. Sonn, Abelian extensions of global fields with constant local degrees, Math. Res. Lett. 13 (4) (2006) 599–607; C.D. Popescu, Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields, J. Number Theory 115 (2005) 27–44] that if F is a global field, then the n-torsion subgroup Brn(F) of its Brauer group Br(F) equals the relative Brauer group Br(Ln/F) of an abelian extension Ln/F, for all n∈Z⩾1. We conjecture that this property characterizes the global fields within the class of infinite fields which are finitely generated over their prime fields. In the first part of this paper, we make a first step towards proving this conjecture. Namely, we show that if F is a non-global infinite field, which is finitely generated over its prime field and ℓ≠char(F) is a prime number such that μℓ2⊆F×, then there does not exist an abelian extension L/F such that Brℓ(F)=Br(L/F). The second and third parts of this paper are concerned with a close analysis of the link between the hypothesis μℓ2⊆F× and the existence of an abelian extension L/F such that Brℓ(F)=Br(L/F), in the case where F is a Henselian valued field.

One of the open questions that has emerged in the study of the projective Schur group PS(F) of a field F is whether or not PS(F) is an algebraic relative Brauer group over F, i.e. does there exist an algebraic extension L/F such that PS(F) = Br(L/F)? We show that the same question for the Schur group of a number field has a negative answer. For the projective Schur group, 110 counterexample is known. In this paper we prove that PS(F) is an algebraic relative Brauer group for all Henselian valued fields F of equal characteristic whose residue field is a local or global field. For this, we first show how PS(F) is determined by PS(k) for an equicharacteristic Henselian field with arbitrary residue field k. (c) 2006 Elsevier B.V. All rights reserved.

We prove that, given a global field K and a positive integer n, there exists an abelian extension L/K (of exponent n) such that the local degree of L/K is equal to n at every finite prime of K, and is equal to two at the real primes if n = 2. As a consequence, we prove that the n-torsion subgroup of the Brauer group of K is equal to the relative Brauer group of L/K.

For any positive integer n n , there exist polynomials f ( x ) ∈ Z [ x ] f(x)\in \mathbb {Z}[x] of degree n n which are irreducible over Q \mathbb {Q} and reducible over Q p \mathbb {Q}_{p} for all primes p p if and only if n n is composite. In fact, this result holds over arbitrary global fields.