Elliptic Gauss sums and applications to point counting

We define a class of algebras over finite fields, called polynomially cyclic algebras, which extend the class of abelian field extensions. We study the structure of these algebras; furthermore, we define and investigate properties of Lagrange resolvents and Gauss and Jacobi sums.Natural examples of polynomially cyclic algebras are for instance algebras of the form Fp[X]/(Fq(X)) where p,q are distinct odd primes and Fq is the cyclotomic polynomial. Further examples occur similarly on replacing the cyclotomic polynomials with factors of division polynomials of elliptic curves. Finally, Gauss and Jacobi sums over polynomially cyclic algebras are applied for improving current algorithms for counting the number of points of elliptic curves over finite fields.