[[Ring theory MOC]]
# Field theory MOC

Fields are a very nice object to have around, but the mathematics of fields themselves can be rather complicated.
In this section of notes, as well as related ones such as [[Algebraic number theory MOC]],
we will use $K$ to refer to a field instead of $\mathbb{K}$, to emphasize that there is no longer one solid “field of discourse”.

Field theory is of course the study of [[Category of fields]],
but since there are no morphisms between fields of differing [[characteristic]],
we might as well consider the subcategories [[Category of fields of characteristic p]].


## Objects

A [[Field]] may be further classified as follows:

- [[Galois field]]
- [[Field of prime characteristic]]
- [[Prime field]]
- [[Perfect field]]
- [[Algebraically closed field]]


## Extensions

As noted in the Zettel for [[Category of fields]], a morphism (unless it’s an isomorphism) of fields is more naturally viewed as a [[Field extension]].

### Types of field extension

- [[Finite field extension]]
- [[Simple extension]]
- [[Finitely generated field extension]]
- [[Automorphism of a field extension]]
- [[Normal extension]]
- [[Separable extension]]
- [[Galois extension]]


### Constructions of extensions

- [[Adjoining a root to a field]]
- [[Algebraic closure]]
- [[Splitting field]]
- [[Fixed field of an automorphism group]]

### Invariants of extensions

- [[Discriminant of a separable extension]]
- [[Automorphism of a field extension]], [[Galois group]]


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