Induced Quaternion Algebras in the Schur Group

Abstract

Let K be a finite, imaginary and abelian extension of the rational number field Q, and let M be the maximal real subfield of K. It is well known that each element of order 2 in S(K), the Schur group of K, is induced from an element of order 2 in B(M), the Brauer group of M; i.e., if D is a quaternion division algebra central over K such that its class [D] in B(K) is in fact in S(K) then [D] = [B ⊗MK] where B is a quaternion division algebra with [B] ∈ B(M). A natural question to ask is: “When is every element of S(K) of order 2 induced from S(M)?” The main result of this paper is to provide necessary and sufficient conditions for this to occur when G(L/K), the Galois group of L over K, is cyclic where L is the smallest root of unity field containing K.