A subfield of a field is a subring which is itself a field. #m/def/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. #m/thm/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.