Existence of finite fields of every prime power order

Statement

Let $p$ be a prime number and let $n \ge 1$. Then there is a finite field of order $q = p^n$.

Remarks

The proof: let $L$ be a splitting field of $f = x^q - x$ over $\mathbb{F}_p$. The derivative $f' = q x^{q-1} - 1 = -1$ has no common root with $f$, so $f$ has $q$ distinct roots in $L$, by the criterion for repeated roots. The roots of $f$ form a subfield of $L$, because $(a + b)^q = a^q + b^q$ and $(a b)^q = a^q b^q$. This subfield has $q$ elements. So for each $n$ there is an irreducible polynomial of degree $n$ over $\mathbb{F}_p$.

History

E. H. Moore first proved the classification of finite fields in 1893: there is a field of each prime power order, and all fields of a given order are isomorphic.

Explanations

L1 human

The field with $p^n$ elements is made of the roots of $x^{p^n} - x$. The proof checks that these roots are distinct and closed under the field operations.