Editing Finite field (section)

== Existence and uniqueness ==
Let <math>q=p^n</math> be a [[prime power]], and <math>F</math> be the [[splitting field]] of the polynomial 
<math display="block">P = X^q-X</math>
over the prime field <math>\mathrm{GF}(p)</math>. This means that <math>F</math> is a finite field of lowest order, in which <math>P</math> has <math>q</math> distinct roots (the [[formal derivative]] of <math>P</math> is <math>P'=-1</math>, implying that <math>\mathrm{gcd}(P,P')=1</math>, which in general implies that the splitting field is a [[separable extension]] of the original). The [[#powersum|above identity]] shows that the sum and the product of two roots of <math>P</math> are roots of <math>P</math>, as well as the multiplicative inverse of a root of <math>P</math>. In other words, the roots of <math>P</math> form a field of order <math>q</math>, which is equal to <math>F</math> by the minimality of the splitting field.

The uniqueness up to isomorphism of splitting fields implies thus that all fields of order <math>q</math> are isomorphic. Also, if a field <math>F</math> has a field of order <math>q=p^k</math> as a subfield, its elements are the <math>q</math> roots of <math>X^q-X</math>, and <math>F</math> cannot contain another subfield of order <math>q</math>.

In summary, we have the following classification theorem first proved in 1893 by [[E.&nbsp;H.&nbsp;Moore]]:<ref name="moore">{{citation|first=E. H.|last=Moore|author-link=E. H. Moore|chapter=A doubly-infinite system of simple groups|editor=E. H. Moore |display-editors=etal |title=Mathematical Papers Read at the International Mathematics Congress Held in Connection with the World's Columbian Exposition|pages=208–242|publisher=Macmillan & Co.|year=1896}}</ref>
<blockquote>
The order of a finite field is a prime power. For every prime power <math>q</math> there are fields of order <math>q</math>, and they are all isomorphic. In these fields, every element satisfies
<math display="block">
 x^q = x,
</math>
and the polynomial <math>X^q - X</math> factors as
<math display="block">
 X^q - X = \prod_{a\in F} (X - a).
</math>
</blockquote>

It follows that <math>\mathrm{GF}(p^n)</math> contains a subfield isomorphic to <math>\mathrm{GF}(p^m)</math> if and only if <math>m</math> is a divisor of <math>n</math>; in that case, this subfield is unique. In fact, the polynomial <math>X^{p^m}-X</math> divides <math>X^{p^n}-X</math> if and only if <math>m</math> is a divisor of <math>n</math>.