Editing Field extension (section)

== Caveats ==
The notation ''L'' / ''K'' is purely formal and does not imply the formation of a [[quotient ring]] or [[quotient group]] or any other kind of division. Instead the slash expresses the word "over". In some literature the notation ''L'':''K'' is used.

It is often desirable to talk about field extensions in situations where the small field is not actually contained in the larger one, but is naturally embedded. For this purpose, one abstractly defines a field extension as an [[injective function|injective]] [[ring homomorphism]] between two fields.
''Every'' non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper [[Ideal_(ring_theory)|ideals]], so field extensions are precisely the [[morphism]]s in the [[category of fields]].

Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.