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 F ⊆ K {\displaystyle F\subseteq K} be a Galois extension with Galois group G {\displaystyle G} . The classical normal basis theorem states that there is an element β ∈ K {\displaystyle \beta \in K} such that { σ ( β ) : σ ∈ G } {\displaystyle \{\sigma (\beta ):\sigma \in G\}} forms a basis of K {\displaystyle K} , considered as a vector space over F {\displaystyle F} . That is, any element α ∈ K {\displaystyle \alpha \in K} can be written uniquely as α = ∑ σ ∈ G a σ σ ( β ) {\textstyle \alpha =\sum _{\sigma \in G}a_{\sigma }\,\sigma (\beta )} for some coefficients a σ ∈ F {\displaystyle a_{\sigma }\in F} . A normal basis contrasts with a primitive element basis of the form { 1 , β , β 2 , … , β n − 1 } {\displaystyle \{1,\beta ,\beta ^{2},\ldots ,\beta ^{n-1}\}} , where β ∈ K {\displaystyle \beta \in K} is an element whose minimal polynomial has degree n = [ K : F ] {\displaystyle n=[K:F]} .
Group representation point of view 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; thus K is also a left module for the group algebra F[G]. Every homomorphism of left F[G]-modules ϕ : F [ G ] → K {\displaystyle \phi :F[G]\rightarrow K} is of form ϕ ( σ ) = σ ( β ) {\displaystyle \phi (\sigma )=\sigma (\beta )} for some β ∈ K {\displaystyle \beta \in K} . Since { σ : σ ∈ G } {\displaystyle \{\sigma :\sigma \in G\}} is a linear basis of F[G] over F, we see that ϕ {\displaystyle \phi } is bijective iff β {\displaystyle \beta } 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 K ≅ F [ G ] {\displaystyle K\cong F[G]} as a left F [ G ] {\displaystyle F[G]} -module: K is isomorphic to the regular representation. Also, a given β {\displaystyle \beta } generates a normal basis iff, when considering K as a G-representation, β {\displaystyle \beta } does not lie in any proper subrepresentation.
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