Vol. 45, No. 2, 1973

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Maximal subfields of tensor products

Burton I. Fein and Murray M. Schacher

Vol. 45 (1973), No. 2, 479–483

Let D1 and D2 be finite-dimensional division rings with center K such that D1 KD2 is a division ring. If L1 and L2 are maximal subfields of D1 and D2, respectively, then clearly L1 KL2 is a maximal subfield of D1 KD2. In this note the converse question is considered: does there exist a maximal subfield L of D1 KD2 which is not isomorphic to L1 KL2 for maximal subfields L1 and L2 of D1 and D2? Examples are given to show that such noncomposite L may fail to exist even when K is a local field. For K an algebraic number field, however, it is shown that infinitely many noncomposite L always exist.

Mathematical Subject Classification
Primary: 16A40
Received: 11 November 1971
Published: 1 April 1973
Burton I. Fein
Murray M. Schacher