# Normal basis

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.

Note that this proof would also apply in the case of a cyclic Kummer extension.

means we have a direct sum of F[G]-modules (by the Chinese remainder theorem):

The normal basis is frequently used in cryptographic applications based on the discrete logarithm problem, such as elliptic curve cryptography, since arithmetic using a normal basis is typically more computationally efficient than using other bases.

If K/F is a Galois extension and x in E generates a normal basis over F, then x is free in K/F. If x has the property that for every subgroup H of the Galois group G, with fixed field KH, x is free for K/KH, then x is said to be completely free in K/F. Every Galois extension has a completely free element.[2]