Editing Algebraic extension (section)

==Generalizations==
{{Main|Substructure (mathematics)}}
[[Model theory]] generalizes the notion of algebraic extension to arbitrary theories: an [[embedding]] of ''M'' into ''N'' is called an '''algebraic extension''' if for every ''x'' in ''N'' there is a [[Well-formed formula|formula]] ''p'' with parameters in ''M'', such that ''p''(''x'') is true and the set

:<math>\left\{y\in N \mid p(y)\right\}</math>

is finite. It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension. The [[Galois group]] of ''N'' over ''M'' can again be defined as the [[Group (mathematics)|group]] of [[automorphism]]s, and it turns out that most of the theory of Galois groups can be developed for the general case.