Let D be a division algebra of finite dimension n2
over it’s center F. Suppose D has an involution, τ, of the first kind, of symplectic type (e.g. , p. 169). By the theory of the pfaffian, τ symmetric elements have degree less than n/2 over F. On the other hand, Tamagawa has shown (unpublished) that involutions like τ are closely related to minimal symmetric idempotents in D ⊗
F D. This author began by examining and trying to generalize these relationships. But before any theory seemed possible for division algebras, a theory relating subfields and symmetric idempotents was required. This investigation gave rise to the results presented here, especially the main theorem in Section Two.