irreducible polynomial

Let  $f(x)=a_0+a_{1}x+\mathrm{\cdots }+a_n{x}^n$  be a polynomial with complex coefficients $a_{\nu }$ and with the degree (http://planetmath.org/Polynomial)  $n>0$.  If $f(x)$ can not be written as product of two polynomials with positive degrees and with coefficients in the field  $\mathbb{Q}(a_0,a_1,\mathrm{\dots },a_n)$,  then the polynomial $f(x)$ is said to be .  Otherwise, $f(x)$ is reducible.

Examples.  All linear polynomials are .  The polynomials $x^2-3$, $x^2+1$ and $x^2-i$ are (although they split in linear factors in the fields $\mathbb{Q}(\sqrt{3})$, $\mathbb{Q}(i)$ and $\mathbb{Q}(\frac{1+i}{\sqrt{2}})$, respectively).  The polynomials $x^4+4$ and $x^6+1$ are not .

The above definition of polynomial is special case of the more general setting where $f(x)$ is a non-constant polynomial in the polynomial ring $K[x]$ of a field $K$; if $f(x)$ is not expressible as product of two polynomials with positive degrees in the ring $K[x]$, then $f(x)$ is (in $K[x]$).

Example.  If $K$ is the Galois field with two elements (0 and 1), then the trinomial $x^2+x+1$ of $K[x]$ is (because an equation  $x^2+x+1=(x+a)(x+b)$  would imply the two conflicting conditions  $a+b=1$  and  $ab=1$).

Title	irreducible polynomial
Canonical name	IrreduciblePolynomial
Date of creation	2013-03-22 14:24:22
Last modified on	2013-03-22 14:24:22
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	18
Author	pahio (2872)
Entry type	Definition
Classification	msc 12D10
Synonym	prime polynomial
Synonym	indivisible polynomial
Related topic	EisensteinCriterion
Related topic	Irreducible
Related topic	Monic2
Defines	irreducible polynomial
Defines	reducible