Let K be a number field of degree d so that \(K/\mathbb {Q}\) is a Galois extension. The normal basis theorem states that K has a \(\mathbb {Q}\) -basis consisting of algebraic conjugates, in fact K contains infinitely many such bases. We prove an effective version of this theorem, obtaining a normal basis for \(K/\mathbb {Q}\) of bounded Weil height with an explicit bound in terms of the degree and discriminant of K. In the case when d is prime, we obtain a particularly good bound using a different method.
