Brauer groups of fields algebraic over Q

Abstract

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;