Editing Irreducible polynomial (section)

== Over the reals ==

Over the [[real number|field of reals]], the [[degree of a polynomial|degree]] of an irreducible univariate polynomial is either one or two. More precisely, the irreducible polynomials are the polynomials of degree one and the [[quadratic polynomial]]s <math>ax^2 + bx + c</math> that have a negative [[discriminant]] <math>b^2 - 4ac.</math> It follows that every non-constant univariate polynomial can be factored as a product of polynomials of degree at most two. For example, <math>x^4 + 1</math> factors over the real numbers as <math>\left(x^2 + \sqrt{2}x + 1\right)\left(x^2 - \sqrt{2}x + 1\right),</math> and it cannot be factored further, as both factors have a negative discriminant: <math>\left(\pm\sqrt{2}\right)^2 - 4 = -2 < 0.</math>