n-Dimensional algebras over a field with a cyclic extension of degree n

We give a geometric method of classifying algebras A _n,K , n-dimensional over a field K, with a cyclic extension of degree n. Algebras A _n,K without zero divisors satisfying some conditions are classified. In particular, we determine all n-dimensional division algebras over a finite field F _q when n is prime and q is large enough.This research was supported in part by a grant from the M U R S T (40 % funds).