# irreducible polynomials over finite field

Theorem.  Over a finite field $F$, there exist irreducible polynomials of any degree.

Proof.  Let $n$ be a positive integer, $p$ be the characteristic of $F$, $\mathbb{F}_{p}$ be the prime subfield, and $p^{r}$ be the order (http://planetmath.org/FiniteField) of the field $F$.  Since $p^{r}\!-\!1$ is a divisor of $p^{rn}\!-\!1$, the zeros of the polynomial $X^{p^{r}}\!-\!X$ form in  $G:=\mathbb{F}_{p^{rn}}$  a subfield isomorphic to $F$.  Thus, one can regard $F$ as a subfield of $G$.  Because

 $[G\!:\!F]=\frac{[G\!:\!\mathbb{F}_{p}]}{[F\!:\!\mathbb{F}_{p}]}=\frac{rn}{r}=n,$

the minimal polynomial of a primitive element of the field extension $G/F$ is an irreducible polynomial of degree $n$ in the ring $F[X].$

Title irreducible polynomials over finite field IrreduciblePolynomialsOverFiniteField 2013-03-22 17:43:14 2013-03-22 17:43:14 pahio (2872) pahio (2872) 7 pahio (2872) Theorem msc 12E20 msc 11T99 FiniteField