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Concept
Normal basis
Summary
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.
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 .
A field extension K / F 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 F[G]. Every homomorphism of left F[G]-modules is of form for some . Since is a linear basis of F[G] 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 K / F 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.
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 β ∈ K such that
is a basis of K over F.
In case the Galois group is cyclic as above, generated by with the normal basis theorem follows from two basic facts.
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