Editing Splitting field (section)

==Properties==
An extension ''L'' that is a [[Algebraic closure#Existence of an algebraic closure and splitting fields|splitting field for a set of polynomials]] ''p''(''X'') over ''K'' is called a [[Field extension#Normal, separable and Galois extensions|normal extension]] of ''K''.

Given an [[algebraically closed field]] ''A'' containing ''K'', there is a unique splitting field ''L'' of ''p'' between ''K'' and ''A'', generated by the roots of ''p''. If ''K'' is a [[field extension|subfield]] of the [[complex number]]s, the existence is immediate. On the other hand, the existence of [[algebraic closure]]s in general is often [[mathematical proof|proved]] by 'passing to the limit' from the splitting field result, which therefore requires an independent proof to avoid [[circular definition|circular reasoning]].

Given a [[separable extension]] ''K''′ of ''K'', a '''Galois closure''' ''L'' of ''K''′ is a type of splitting field, and also a [[Galois extension]] of ''K'' containing ''K''′ that is minimal, in an obvious sense. Such a Galois closure should contain a splitting field for all the polynomials ''p'' over ''K'' that are [[Minimal polynomial (field theory)|minimal polynomials]] over ''K'' of elements of ''K''′.