Cyclotomic Splitting Fields

Mollin, R. A.

Canadian mathematical bulletin, Tome 25 (1982) no. 2, pp. 222-229

Résumé

Let D be a division algebra whose class [D] is in B(K), the Brauer group of an algebraic number field K. If [D⊗K L] is the trivial class in B(L), then we say that L is a splitting field for D or L splits D. The splitting fields in D of smallest dimension are the maximal subfields of D. Although there are infinitely many maximal subfields of D which are cyclic extensions of K; from the perspective of the Schur Subgroup S(K) of B(K) the natural splitting fields are the cyclotomic ones. In (Cyclotomic Splitting Fields, Proc. Amer. Math. Soc. 25 (1970), 630-633) there are errors which have led to the main result of this paper, namely to provide necessary and sufficient conditions for (D) in S(K) to have a maximal subfield which is a cyclic cyclotomic extension of K, a finite abelian extension of Q. A similar result is provided for quaternion division algebras in B(K).

Détail

DOI : 10.4153/CMB-1982-031-3

Mots-clés : 16A26, 16A65, Cyclotomic field, splitting field, division algebra, maximal subfield