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. { β , Φ ( β ) , Φ 2 ( β ) , … , Φ n − 1 ( β ) }   =   { β , β q , β q 2 , … , β q n − 1 } {displaystyle {eta ,Phi (eta ),Phi ^{2}(eta ),ldots ,Phi ^{n-1}(eta )} = {eta ,eta ^{q},eta ^{q^{2}},ldots ,eta ^{q^{n-1}}!}} K   ≅   F [ X ] / ( X n − 1 ) {displaystyle K cong F/(X^{n}{-},1)} X n − 1   =   X 3 − 1   =   ( X + 1 ) ( X 2 + X + 1 )   ∈   F [ X ] {displaystyle X^{n}-1 = X^{3}-1 = (X{+}1)(X^{2}{+}X{+}1) in F} K   ≅   F [ X ] ( X 3 − 1 )   ≅   F [ X ] ( X + 1 ) ⊕ F [ X ] ( X 2 + X + 1 ) . {displaystyle K cong {frac {F}{(X^{3}{-},1)}} cong {frac {F}{(X{+}1)}}oplus {frac {F}{(X^{2}{+}X{+}1)}}.} t 2 3 − t   =   t ( t + 1 ) ( t 3 + t + 1 ) ( t 3 + t 2 + 1 )   ∈   F [ t ] , {displaystyle t^{2^{3}}-t = t(t{+}1)(t^{3}{+}t{+}1)(t^{3}{+}t^{2}{+}1) in F,} L   ≅   F 2 [ X ] / ( X 4 − 1 )   =   F 2 [ X ] / ( X + 1 ) 4 . {displaystyle L cong mathbb {F} _{2}/(X^{4}{-}1) = mathbb {F} _{2}/(X{+}1)^{4}.} α   =   ( a 2 , a 1 , a 0 )   =   a 2 Φ 2 ( β ) + a 1 Φ ( β ) + a 0 β   =   a 2 β 4 + a 1 β 2 + a 0 β , {displaystyle alpha = (a_{2},a_{1},a_{0}) = a_{2}Phi ^{2}(eta )+a_{1}Phi (eta )+a_{0}eta = a_{2}eta ^{4}+a_{1}eta ^{2}+a_{0}eta ,} 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 F ⊂ K {displaystyle Fsubset 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 eta in K} such that { g ( β )   for   g ∈ G } {displaystyle {g(eta ) { extrm {for}} gin G}} forms a basis of K, considered as a vector space over F. That is, any element α ∈ K {displaystyle alpha in K} can be written uniquely as α = ∑ g ∈ G a g g ( β ) {displaystyle extstyle alpha =sum _{gin G}a_{g},g(eta )} for coefficients a g ∈ F . {displaystyle a_{g}in F.} A normal basis contrasts with a primitive element basis of the form { 1 , β , β 2 , … , β n − 1 } {displaystyle {1,eta ,eta ^{2},ldots ,eta ^{n-1}}} , where β ∈ K {displaystyle eta in K} is an element whose minimal polynomial has degree n = [ K : F ] {displaystyle n=} . For finite fields this can be stated as follows: Let F = G F ( q ) = F q {displaystyle F=GF(q)=mathbb {F} _{q}} denote the field of q elements, where q = pm is a prime power, and let K = G F ( q n ) = F q n {displaystyle K=GF(q^{n})=mathbb {F} _{q^{n}}} denote its extension field of degree n ≥ 1. Here the Galois group is G = Gal ( K / F ) = { 1 , Φ , Φ 2 , … , Φ n − 1 } {displaystyle G={ ext{Gal}}(K/F)={1,Phi ,Phi ^{2},ldots ,Phi ^{n-1}}} with Φ n = 1 , {displaystyle Phi ^{n}=1,} a cyclic group generated by the relative Frobenius automorphism Φ ( α ) = α q , {displaystyle Phi (alpha )=alpha ^{q},} with Φ n = 1 = Id K . {displaystyle Phi ^{n}=1={ extrm {Id}}_{K}.} 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 Φ {displaystyle Phi } with Φ n = 1 , {displaystyle Phi ^{n}=1,} 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 χ ( h 1 h 2 ) = χ ( h 1 ) χ ( h 2 ) {displaystyle chi (h_{1}h_{2})=chi (h_{1})chi (h_{2})} ; then any distinct characters χ 1 , χ 2 , … {displaystyle chi _{1},chi _{2},ldots } are linearly independent in the K-vector space of mappings. We apply this to the Galois group automorphisms χ i = Φ i : K → K , {displaystyle chi _{i}=Phi ^{i}:K o K,} thought of as mappings from the multiplicative group H = K × {displaystyle H=K^{ imes }} . Now K ≅ F n {displaystyle Kcong F^{n}} as an F-vector space, so we may consider Φ : F n → F n {displaystyle Phi :F^{n} o F^{n}} as an element of the matrix algebra M n ( F ) ; {displaystyle M_{n}(F);} since its powers 1 , Φ , … , Φ n − 1 {displaystyle 1,Phi ,ldots ,Phi ^{n-1}} are linearly independent (over K and a fortiori over F), its minimal polynomial must have degree at least n, i.e. it must be X n − 1 {displaystyle X^{n}-1} . We conclude that the group algebra of G is F [ G ] ≅ F [ X ] / ( X n − 1 ) , {displaystyle Fcong F/(X^{n}{-},1),} a quotient of the polynomial ring F, and the F-vector space K is a module (or representation) for this algebra. The second basic fact is the classification of modules over a PID such as F. These are just direct sums of cyclic modules of the form F [ X ] / ( f ( x ) ) , {displaystyle F/(f(x)),} where f(x) must be divisible by Xn − 1. (Here G acts by Φ ⋅ X i = X i + 1 . {displaystyle Phi cdot X^{i}=X^{i+1}.} ) But since dim F ⁡ F [ X ] / ( X n − 1 ) = dim F ⁡ ( K ) = n , {displaystyle dim _{F}F/(X^{n}{-},1)=dim _{F}(K)=n,} we can only have f(x) = Xn − 1, and as F-modules, namely the regular representation of G. (Note this is not an isomorphism of rings or F-algebras!) Now the basis { 1 , X , X 2 , … , X n − 1 } {displaystyle {1,X,X^{2},ldots ,X^{n-1}}} on the right side of this isomorphism corresponds to a normal basis { β , Φ ( β ) , Φ 2 ( β ) , … , Φ m − 1 ( β ) } {displaystyle {eta ,Phi (eta ),Phi ^{2}(eta ),ldots ,Phi ^{m-1}(eta )}} of K on the left. Note that this proof would also apply in the case of a cyclic Kummer extension.