Editing Normal basis (section)

=== Proof for finite fields ===
In case the Galois group is cyclic as above, generated by <math>\Phi</math> with <math>\Phi^n=1,</math> 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 <math>\chi(h_1h_2)=\chi(h_1)\chi(h_2)</math>; then any distinct characters <math>\chi_1,\chi_2,\ldots </math> are linearly independent in the ''K''-vector space of mappings. We apply this to the Galois group automorphisms <math>\chi_i=\Phi^i: K \to K,</math> thought of as mappings from the multiplicative group <math>H=K^\times</math>. Now  <math>K\cong F^n</math>as an ''F''-vector space, so we may consider <math>\Phi : F^n\to F^n</math> as an element of the matrix algebra M<sub>''n''</sub>(''F''); since its powers <math>1,\Phi,\ldots,\Phi^{n-1}</math> are linearly independent (over ''K'' and a fortiori over ''F''), its [[minimal polynomial (field theory)|minimal polynomial]] must have degree at least ''n'', i.e. it must be <math>X^n-1</math>.

The second basic fact is the classification of finitely generated [[Modules over a pid|modules over a PID]] such as <math>F[X]</math>. Every such module ''M'' can be represented as <math display="inline">M \cong\bigoplus_{i=1}^k \frac{F[X]}{(f_i(X))}</math>, where <math>f_i(X)</math> may be chosen so that they are monic polynomials or zero and <math>f_{i+1}(X)</math> is a multiple of <math>f_i(X)</math>. <math>f_k(X)</math> is the monic polynomial of smallest degree annihilating the module, or zero if no such non-zero polynomial exists. In the first case <math display="inline">\dim_F M = \sum_{i=1}^k \deg f_i</math>, in the second case <math>\dim_F M = \infty</math>. In our case of cyclic ''G'' of size ''n'' generated by <math>\Phi</math> we have an ''F''-algebra isomorphism <math display="inline">F[G]\cong \frac {F[X]}{(X^n-1)}</math> where ''X'' corresponds to <math>1 \cdot \Phi</math>, so every <math>F[G]</math>-module may be viewed as an <math>F[X]</math>-module with multiplication by ''X'' being multiplication by <math>1\cdot\Phi</math>. In case of ''K'' this means <math>X\alpha = \Phi(\alpha)</math>, so the monic polynomial of smallest degree annihilating ''K'' is the minimal polynomial of <math>\Phi</math>. Since ''K'' is a finite dimensional ''F''-space, the representation above is possible with <math>f_k(X)=X^n-1</math>. Since <math>\dim_F(K) = n,</math> we can only have <math>k=1</math>, and <math display="inline">K \cong \frac{F[X]}{(X^n{-}\,1)}</math> as ''F''[''X'']-modules. (Note this is  an isomorphism of ''F''-linear spaces, but ''not'' of rings or ''F''-algebras.) This gives isomorphism of <math>F[G]</math>-modules <math>K\cong F[G]</math> that we talked about above, and under it the basis <math>\{1,X,X^2,\ldots,X^{n-1}\}</math> on the right side corresponds to a normal basis <math>\{\beta, \Phi(\beta),\Phi^2(\beta),\ldots,\Phi^{n-1}(\beta)\}</math> of ''K'' on the left.

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