## You are here

Homeirreducible polynomials over finite field

## Primary tabs

# irreducible polynomials over finite field

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

Proof.β 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 of the field $F$.β Since $p^{r}\!-\!1$ is a divisor of $p^{{rn}}\!-\!1$, the zeros of the polynomial $X^{{p^{r}}}\!-\!X$ form inβ $G:=\mathbb{F}_{{p^{{rn}}}}$β a subfield isomorphic to $F$.β Thus, one can regard $F$ as a subfield of $G$.β 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 $F[X].$

Related:

FiniteField

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

12E20*no label found*11T99

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

8:30 pm

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier