In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number theory, the study of the more refined question of the existence of a normal integral basis is part of Galois module theory. Normal basis theorem Let be a Galois extension with Galois group . The classical normal basis theorem states that there is an element such that forms a basis of K, considered as a vector space over F. That is, any element can be written uniquely as for some elements A normal basis contrasts with a primitive element basis of the form , where is an element whose minimal polynomial has degree . Group representation point of view A field extension with Galois group G can be naturally viewed as a representation of the group G over the field F in which each automorphism is represented by itself. Representations of G over the field F can be viewed as left modules for the group algebra . Every homomorphism of left -modules is of form for some . Since is a linear basis of over F, it follows easily that is bijective iff generates a normal basis of K over F. The normal basis theorem therefore amounts to the statement saying that if is finite Galois extension, then as left -module. In terms of representations of G over F, this means that K is isomorphic to the regular representation. Case of finite fields For finite fields this can be stated as follows: Let denote the field of q elements, where q = pm is a prime power, and let denote its extension field of degree n ≥ 1. Here the Galois group is with a cyclic group generated by the q-power Frobenius automorphism with Then there exists an element such that is a basis of K over F. Proof for finite fields In case the Galois group is cyclic as above, generated by with the Normal Basis Theorem follows from two basic facts. The first is the linear independence of characters: a multiplicative character is a mapping χ from a group H to a field K satisfying ; then any distinct characters are linearly independent in the K-vector