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.
