Let
F be a
field. A subset
k of
F is a
subfield of
F if it is a
subring of
F and is itself
a field. Equivalently a subset
k of
F 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.