We investigate the field of rational invariants of the linear action of a finite abelian group in the non modular case. By diagonalization, the group is accurately described by an integer matrix of exponents. We make use of linear algebra to compute a minimal generating set of invariants and the substitution to rewrite any invariant in terms of this generating set. We show that the generating set can be chosen to consist of polynomial invariants. As an application, we provide a symmetry reduction scheme for polynomial systems the solution set of which are invariant by the group action. We furthermore provide an algorithm to find such symmetries given a polynomial system.