Editing Splitting field (section)

==Definition==
A '''splitting field''' of a polynomial ''p''(''X'') over a field ''K'' is a field extension ''L'' of ''K'' over which ''p'' factors into linear factors

:<math>p(X) = c \prod_{i=1}^{\deg p} (X - a_i)</math> 

where <math>c \in K</math> and for each <math>i</math> we have <math>X - a_i \in L[X]</math> with ''a<sub>i</sub>'' not necessarily distinct and such that the [[root of a polynomial|roots]] ''a<sub>i</sub>'' generate ''L'' over ''K''. The extension ''L'' is then an extension of minimal [[Degree of a field extension|degree]] over ''K'' in which ''p'' splits. It can be shown that such splitting fields exist and are unique [[up to]] [[isomorphism]]. The amount of freedom in that isomorphism is known as the [[Galois group]] of ''p'' (if we assume it is [[separable polynomial|separable]]).

A splitting field of a set ''P'' of polynomials is the smallest field over which each of the polynomials in ''P'' splits.