# 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 $F$, ${\mathrm{\pi \x9d\x94\xbd}}_{p}$ be the prime subfield^{}, and ${p}^{r}$ be the order (http://planetmath.org/FiniteField) of the field $F$.β Since ${p}^{r}-1$ is a divisor^{} of ${p}^{r\beta \x81\u2019n}-1$, the zeros of the polynomial^{} ${X}^{{p}^{r}}-X$ form inβ $G:={\mathrm{\pi \x9d\x94\xbd}}_{{p}^{r\beta \x81\u2019n}}$β a subfield^{} isomorphic to $F$.β Thus, one can regard $F$ as a subfield of $G$.β Because

$$[G:F]=\frac{[G:{\mathrm{\pi \x9d\x94\xbd}}_{p}]}{[F:{\mathrm{\pi \x9d\x94\xbd}}_{p}]}=\frac{r\beta \x81\u2019n}{r}=n,$$ |

the minimal polynomial of a primitive element^{} of the field extension $G/F$ is an irreducible polynomial of degree $n$ in the ring $F\beta \x81\u2019[X].$

Title | irreducible polynomials over finite field |
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Canonical name | IrreduciblePolynomialsOverFiniteField |

Date of creation | 2013-03-22 17:43:14 |

Last modified on | 2013-03-22 17:43:14 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 12E20 |

Classification | msc 11T99 |

Related topic | FiniteField |