Decomposition of algebras

Abstract

In this paper we deal with the problem of decomposing finite communtative Q-algebras as a direct product of local Q-algebras. We solve this problem by reducing it to the problem of finding a decomposition of finite algebras over finite field. We will show that it is possible to define a lifting process that allows to reconstruct the answer over the rational numbers. This lifting appears to be very efficient since it is a quadratic lifting that doesn't require stepwise inversions. It is easy to see that the Berlekamp-Hensel algorithm for the factorization of polynomials is a special case of this argument.