In this paper we give a deterministic construction for a Normal basis of Kf over K, where K' is an abelian extension of finite degree n over some commutative, infinite field K. No particular property for K is needed.We use polynomials already defined in [2], [5, chapter 5], In these papers the field K was finite. These polynomials were used in order to construct SDMC codes [5,chapter 6], to determine bivariate codes [2],[3], or to obtain the best possible principal nilpotent codes [6], Our construction is deduced from new properties of these polynomials