p-Algebras

A p-algebra is a central simple algebra A over a field of characteristic p>0 such that [A]∈Brp(F), the p-th component of Br(F), that is, the exponent of A is a power of p (equivalently, the index is a power of p). The structure theory of these algebras was developed by Albert in the thirties and was presented in an improved form in Chapter VII of his Structure of Algebras (see also Teichmüller [36]). The culminating result of Albert’s theory is that, in his terminology, any p-algebra is cyclically representable, that is, is similar to a cyclic algebra. This is proved in two stages: the first, in which it is shown that any p-algebra is similar to a tensor product of cyclic algebras and the second, in which it is proved that the tensor product of two cyclic p-algebras is cyclic. The first of these results is proved in Section 4.2. Purely inseparable splitting fields play an important role in the theory. We recall that K/F is purely inseparable if K/F is algebraic and its subfield of separable elements coincides with F, or equivalently, for any a∈K there exists an e≥0 such that $a^{p^{e}}\in F$ . The exponent of K/F is e