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 to refer to a field instead of , 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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