Semidirect product division algebras

Suppose D is a division algebra of degree p over its center F, which contains a primitive p-root of 1. Also suppose D has a maximal separable subfield over F whose Galois group is the semidirect product of the cyclic groups C _p C _q, where q=2, 3, 4, or 6 and is relatively prime to p (In particular this is the case when p is prime ≤7 and D has a maximal separable subfield whose Galois group is solvable.) Then D is cyclic. The proof involves developing a theory of a wider class of algebras, which we call accessible, and proving that they are cyclic.