Field theory MOC

Subfield

A subfield of a field is a subring which is itself a field. field Often we consider these in terms of field extensions.

Tests for subfields

Theorem. Let . Iff for all , we have and implies , then is a subfield. field

Proof

By Subrng test, we have a subrng. Since , we have a subring. Commutativity of implies commutativity of , and every nonzero element has a unit. Therefore is a field.

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