irreducible polynomials over finite field

Theorem.  Over a finite fieldMathworldPlanetmath F, there exist irreducible polynomialsMathworldPlanetmath of any degree.

Proof.  Let n be a positive integer, p be the characteristic of F, 𝔽p be the prime subfieldMathworldPlanetmath, and pr be the order ( of the field F.  Since pr-1 is a divisorMathworldPlanetmathPlanetmath of pr⁒n-1, the zeros of the polynomialMathworldPlanetmathPlanetmathPlanetmath Xpr-X form in  G:=𝔽pr⁒n  a subfieldMathworldPlanetmath isomorphic to F.  Thus, one can regard F as a subfield of G.  Because


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

Title irreducible polynomials over finite field
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