Editing Irreducible polynomial (section)

==Over the complex numbers==

Over the [[complex field]], and, more generally, over an [[algebraically closed field]], a [[univariate polynomial]] is irreducible if and only if its [[degree of a polynomial|degree]] is one. This fact is known as the [[fundamental theorem of algebra]] in the case of the complex numbers and, in general, as the condition of being algebraically closed.

It follows that every nonconstant univariate polynomial can be factored as

:<math>a\left(x - z_1\right) \cdots \left(x - z_n\right)</math>

where <math>n</math> is the degree, <math>a</math> is the leading coefficient and <math>z_1, \dots, z_n</math> are the zeros of the polynomial (not necessarily distinct, and not necessarily having explicit [[algebraic expression]]s).

There are irreducible [[multivariate polynomial]]s of every degree over the complex numbers. For example, the polynomial 
:<math>x^n + y^n - 1,</math> 

which defines a [[Fermat curve]], is irreducible for every positive ''n''.