be a field
. A subset k
if it is a subring
and is itself
a field. Equivalently a subset k
is a subfield
if and only if
1F in k
a-b in k, for all a,b in k
ab in k, for all a,b in k
a-1 in k, for all nonzero a in k.
For example, the field of rational numbers is a subfield of the field
of real numbers which is itself a subfield of the complex numbers.
A finite field has the the integers mod p as a subfield.
See also field extension.