Let and be finite extensions of a field contained in a field . Prove that the -algebra is a field if and only if .
First, define by . Clearly is -bilinear, and so induces an -module homomorphism . In fact is an -algebra homomorphism. Using Proposition 21 in D&F, if and are bases of and over , then spans over . In particular, is surjective.
Suppose is a field. Now is an ideal of , and so must be trivial- so is an isomorphism of -algebras, and thus an isomorphism of fields. Using Proposition 21 on page 421 of D&F, has dimension , and so as desired.
Conversely, suppose . That is, and have the same dimension as -algebras. By Corollary 9 on page 413 of D&F, is injective, and so and are isomorphic as -algebras, hence as rings, and so is a field.