Burton Fein

Brauer-Hilbertian fields

Let F be a field of characteristic p (p = 0 allowed), and let F(t) be the rational function field in one variable over F. We say F is Brauer-Hilbertian if the following holds. For every α in the Brauer group Br(F(t)) of exponent prime to p, there are infinitely many specializations t → a ∈ F such that the specialization α ∈ Br(F) is defined and has exponent equal to that of α. We show every global field is Brauer-Hilbertian, and if K is Hilbertian and F is finite separable over K(t), F is Brauer-Hilbertian

Embedding finite groups in rational division algebras. II

In [7] and [8] the problem of determining the finite subgroups of division rings with specified centers was investigated. In particular, we were concerned with determ~ng for which fields K every odd order finite subgroup of a finite-dimensional division ring central over K is necessarily cyclic. This problem was completely settled for K a local field in [8]. In this paper we answer this question when K is an algebraic number field. Our main result is the following: Let K be an algebraic number field. Then there exists a division ring D finite dimensional and central over K such that D* has a noncyclic subgroup of odd order if and only if K contains a primitive q-th root of unity for some odd prime q. We maintain the notation and terminology of [7J and [8]. Recall that for K a field a K-division ring D is a finite-dimensional central division algebra over K. The dimension of D over K will be denoted by ED: fcl; we use the same notation for dimension of field extensions. We say that D is E-adequate if there is an E-division ring containing I). If a finite group G is contained in the multiplicative group of a K-division ring, we say that G is K-adequate. G is a K-adequate if and only if V(G) is K-adequate where V(G) is the minimal division ring containing G; the structure of T(G) was determined by Amitsur in [2]. By an A-group we will mean a noncyclic odd order group which is a subgroup of some division ring.

On the irreducibility of the iterates of x^n - b

Let K be a field and suppose that f(x) = x n - b is irreducible in K[x]. We discuss the following question: under what conditions are all iterates of f irreducible over K?.

Brauer groups of fields algebraic over Q

In this paper we will be concerned with division rings that are finite dimensional and central over a field K which is an algebraic (possibly infinite dimensional) extension of the rational field Q. In Section 1 we determine necessary and sufficient conditions for an abelian group to occur as the Brauer group of such a field. It should be emphasized that there is little difficulty in showing that our Brauer groups satisfy the requisite properties; the problem is in showing that every group satisfying these properties actually occurs as the Brauer group of some field algebraic over Q. Section 2 is devoted to the proof of a stability property of the Brauer group, one which is preserved under finite extensions. In Section 3 we investigate which of the theorems of [3] and [6] fail in the case where K/Q is infinite dimensional. Many of our theorems hold with only slight modification when the field K is assumed to be algebraic over the function field Zp(t) for some prime p; when this is the case we will mention the relevant result with only an indication of the proof. Thus all fields considered will be algebraic extensions of Q. The notation and terminology of [3, 41 will be in force throughout this paper. In particular, a division ring D which is finite dimensional and central over a field K will be called a K-division ring. If D is a K-division ring with K/Q algebraic, it is well known that the index and exponent of D are equal;

The ordinary quaternions over a Pythagorean field

Let L be a proper finite Galois extension of a field K and let D be a division algebra with center K. If every subfield of D properly containing K contains a K-isomorphic copy of L, it is shown that K must be Pythagorean, L K(V - 1 ), and D is the ordinary quaternions over K. If one assumes only that every maximal subfield of D contains a K isomorphic copy of L, then, under the assumption that (D: K) is finite, it is shown that K is Pythagorean, L = K(V - 1), and D contains the ordinary quaternions over K.

Finite subgroups occuring in finite-dimensional division algebras

In [+6] we proved Conjecture A if K is either an algebraic number field or the completion of an algebraic number field. The proofs given involved extremely technical number theoretic methods and were not applicable to other fields. In this paper we approach this question from a different viewpoint and prove a result valid for arbitrary fields. In the process we prove Conjecture A for a more general class of both K and D, and we obtain greatly simplified proofs of some of the main results of [4-61. By a K-division ring we mean a finite dimensional division algebra over K with center K. In view of Herstein’s result that finite subgroups of division rings of prime characteristic are cyclic [7], we restrict our attention to fields of characteristic zero. We let Q denote the field of rational numbers and, for G a finite multiplicative subgroup of D, we set 9(G) = {C a& ) ai E Q, Ai E G). g’(G) is a finite dimensional division algebra over Q (not necessarily central over Q). Amitsur in [2] determined the structure of g(G); in particular 93(G) depends on G up to isomorphism, and not on D. We denote the center of 93(G) by 8. Since D 3 g(G), D contains the subalgebra generated by K and g(G), which is easily seen to be KC? @ 99(G). We set A(G) = KC? OS S(G). We will maintain this notation throughout this paper.

Sums of corestrictions of cyclic algebras

By a cyclic layer of a finite Galois extension,E/K, of fields one means a cyclic extension,L/F, of fields whereE⊇L⊇F⊇K. LetC(E/K) denote the subgroup of the relative Brauer group, Br(E/K), generated by the various subgroups cor(Br(L/F)) asL/F ranges over all cyclic layers ofE/K and where cor denotes the corestriction map into Br(E/K). We show that forK global, [Br(E/K) :C(E/K)]<∞ and we produce examples whereC(E/K)≠Br(E/K).

RELATIVE BRAUER GROUPS OF DISCRETE VALUED FIELDS

Let E be a non-trivial finite Galois extension of a field K. In this paper we investigate the role that valuation-theoretic properties of E/K play in determining the non-triviality of the relative Brauer group, Br(E/K), of E over K. In particular, we show that when K is finitely generated of transcendence degree 1 over a p-adic field k and q is a prime dividing [E : K], then the following conditions are equivalent: (i) the q-primary component, Br(E/K)q, is non-trivial, (ii) Br(E/K)q is infinite, and (iii) there exists a valuation π of E trivial on k such that q divides the order of the decomposition group of E/K at π.

Brauer groups of algebraic function fields

Let E be a countable field. Then B(E),, the p-primary component of the Brauer group of E, if a countable abelian torsion group. As such, the Ulm invariants of B(E), classify that group modulo its maximal divisible subgroup. For an exposition of the Ulm theory, see [I 11. In [9] (see also [7]) the authors and Jack Sonn determine the Ulm invariants of B(E), when E is a rational function field over a global field. The present paper is motivated by this result, and is concerned with two further lines of investigation suggested by this work. In Section 2 we attempt to extend the results of [9] to the case when E is an algebraic function field over a global field. This case seems considerably more difficult than the rational function field case. If E is a rational function field, B(E), is determined by means of the AuslanderBrumer-Faddeev Theorem [ 7, p. 5 1 ] in terms of Brauer groups and character groups of fields of lower transcendence degree. No such result is available in the algebraic function field case. Nevertheless, we are able to completely determine the Ulm invariants of B(E), at finite ordinals when E is an algebraic function field over a global field; this result is obtained using the Merkurjev-Suslin Theorem [14], Saltman’s theory of generic Galois extensions [ 161, and the results of [8]. For E as above we have very little information about the Ulm invariants at infinite ordinals. We do, however,

Specializations of Brauer classes over algebraic function fields

Let F be either a number field or a field finitely generated of transcendence degree ≥ 1 over a Hilbertian field of characteristic 0, let F (t) be the rational function field in one variable over F , and let α ∈ Br(F (t)). It is known that there exist infinitely many a ∈ F such that the specialization t→ a induces a specialization α→ α ∈ Br(F ), where α has exponent equal to that of α. Now let K be a finite extension of F (t) and let β = resK/F (t)(α). We give sufficient conditions on α and K for there to exist infinitely many a ∈ F such that the specialization t → a has an extension to K inducing a specialization β → β ∈ Br(K), K the residue field of K, where β has exponent equal to that of β. We also give examples to show that, in general, such a ∈ F