Murray Schacher

The admissibility of A5

Abstract Examples are constructed of division rings finite-dimensional and central over the rational field Q having subfields with Galois group isomorphic to the alternating group A5. Similar results are obtained for any number field k when √−1 ∉ k.

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.

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;

Central extensions of symmetric groups as Galois groups

’ Supported in part by the Wolf Foundation. *Supported in part by NSF Grant DMS85-00929. During this work, Murray Schacher was also a Fulbright-Hayes Fellow in Belgium under the auspices of the Commission for Educational Exchange Between the United States, Belgium, and Luxembourg.

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.

Irreducible polynomials which are locally reducible everywhere

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

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).

Solutions of pure equations in rational division algebras, II

Let α be a rational number and m a positive integer. In this paper it is determined which of the pure equations xm + α = 0 have a solution in a finite-dimensional division algebra with center the rational field Q. Necessary and sufficient conditions on m and α are given.