Editing Finite field (section)

== Frobenius automorphism and Galois theory ==

In this section, <math>p</math> is a prime number, and <math>q=p^n</math> is a power of <math>p</math>.

In <math>\mathrm{GF}(q)</math>, the identity {{math|1=(''x'' + ''y'')<sup>''p''</sup> = ''x<sup>p</sup>'' + ''y<sup>p</sup>''}} implies that the map
<math display="block"> \varphi:x \mapsto x^p</math>
is a <math>\mathrm{GF}(p)</math>-[[linear map|linear endomorphism]] and a [[field automorphism]] of <math>\mathrm{GF}(q)</math>, which fixes every element of the subfield <math>\mathrm{GF}(p)</math>. It is called the [[Frobenius automorphism]], after [[Ferdinand Georg Frobenius]].

Denoting by {{math|''φ<sup>k</sup>''}} the [[function composition|composition]] of {{math|''φ''}} with itself {{math|''k''}} times, we have 
<math display="block"> \varphi^k:x \mapsto x^{p^k}.</math>
It has been shown in the preceding section that {{math|''φ''<sup>''n''</sup>}} is the identity. For {{math|0 < ''k'' < ''n''}}, the automorphism {{math|''φ''<sup>''k''</sup>}} is not the identity, as, otherwise, the polynomial 
<math display="block">X^{p^k}-X</math>
would have more than {{math|''p<sup>k</sup>''}} roots.

There are no other {{math|GF(''p'')}}-automorphisms of {{math|GF(''q'')}}. In other words, {{math|GF(''p<sup>n</sup>'')}} has exactly {{math|''n''}} {{math|GF(''p'')}}-automorphisms, which are 
<math display="block">\mathrm{Id}=\varphi^0, \varphi, \varphi^2, \ldots, \varphi^{n-1}.</math>

In terms of [[Galois theory]], this means that {{math|GF(''p''<sup>''n''</sup>)}} is a [[Galois extension]] of {{math|GF(''p'')}}, which has a [[cyclic group|cyclic]] Galois group.

The fact that the Frobenius map is surjective implies that every finite field is [[perfect field|perfect]].