# 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 *K*^{H}, *x* is free for *K*/*K*^{H}, then *x* is said to be **completely free** in *K*/*F*. Every Galois extension has a completely free element.^{[2]}