Editing Irreducible polynomial (section)

==Definition==

If ''F'' is a field, a non-constant polynomial is '''irreducible over ''F''''' if its coefficients belong to ''F'' and it cannot be factored into the product of two non-constant polynomials with coefficients in ''F''. 

A polynomial with integer coefficients, or, more generally, with coefficients in a [[unique factorization domain]] ''R'', is sometimes said to be ''irreducible'' (or ''irreducible over R'') if it is an [[irreducible element]] of the [[polynomial ring]], that is, it is not [[unit (ring theory)|invertible]], not zero, and cannot be factored into the product of two non-invertible polynomials with coefficients in ''R''. This definition generalizes the definition given for the case of coefficients in a field, because, over a field, the non-constant polynomials are exactly the polynomials that are non-invertible and non-zero.

Another definition is frequently used, saying that a polynomial is ''irreducible over R'' if it is irreducible over the [[field of fractions]] of ''R'' (the field of [[rational number]]s, if ''R'' is the integers). This second definition is not used in this article.  The equivalence of the two definitions depends on ''R''.