Editing Normal extension (section)

== Equivalent conditions for normality ==

Let <math>L/K</math> be algebraic. The field ''L'' is a normal extension if and only if any of the equivalent conditions below hold.

* The [[Minimal polynomial (field theory)|minimal polynomial]] over ''K'' of every element in ''L'' splits in ''L'';
* There is a set <math>S \subseteq K[x]</math> of polynomials that each splits over ''L'', such that if <math>K\subseteq F\subsetneq L</math> are fields, then ''S'' has a polynomial that does not split in ''F'';
* All homomorphisms <math>L \to \bar{K}</math> that fix all elements of ''K'' have the same image;
* The group of automorphisms, <math>\text{Aut}(L/K),</math> of ''L'' that fix all elements of ''K'', acts transitively on the set of homomorphisms <math>L \to \bar{K}</math> that fix all elements of ''K''.