Editing Irreducible polynomial (section)

==Field extension==
{{main|Algebraic extension}}
The notions of irreducible polynomial and of [[algebraic field extension]] are strongly related, in the following way.

Let ''x'' be an element of an [[field extension|extension]] ''L'' of a field ''K''. This element is said to be ''algebraic'' if it is a [[zero of a function|root]] of a nonzero polynomial with coefficients in ''K''. Among the polynomials of which ''x'' is a root, there is exactly one which is [[monic polynomial|monic]] and of minimal degree, called the [[minimal polynomial (field theory)|minimal polynomial]] of ''x''. The minimal polynomial of an algebraic element ''x'' of ''L'' is irreducible, and is the unique monic irreducible polynomial of which ''x'' is a root. The minimal polynomial of ''x'' divides every polynomial which has ''x'' as a root (this is [[Abel's irreducibility theorem]]). 

Conversely, if <math>P(X) \in K[X]</math> is a univariate polynomial over a field ''K'', let <math>L = K[X]/P(X)</math> be the [[quotient ring]] of the polynomial ring <math>K[X]</math> by the [[ideal (ring theory)#Ideal generated by a set|ideal generated]] by {{math|''P''}}. Then {{math|''L''}} is a field if and only if {{math|''P''}} is irreducible over {{math|''K''}}. In this case, if {{math|''x''}} is the image of {{math|''X''}} in {{math|''L''}}, the minimal polynomial of {{math|''x''}} is the quotient of {{math|''P''}} by its [[leading coefficient]].

An example of the above is the standard definition of the [[complex number]]s as <math>\mathbb{C} = \mathbb{R}[X]\;/\left(X^2 + 1\right).</math>

If a polynomial {{math|''P''}} has an irreducible factor {{math|''Q''}} over {{math|''K''}}, which has a degree greater than one, one may apply to {{math|''Q''}} the preceding construction of an algebraic extension, to get an extension in which {{math|''P''}} has at least one more root than in {{math|''K''}}. Iterating this construction, one gets eventually a field over which {{math|''P''}} factors into linear factors. This field, unique [[up to]] a [[ring isomorphism|field isomorphism]], is called the [[splitting field]] of {{math|''P''}}.