Examples
We come back to the examples of the definition in Section 7.3.
Usual arithmetic
|
Modular arithmetic
|
|
Rational fields
|
Residue class fields
|
|
The Gaussian integers
|
Z is not a field: most of its elements are not invertible.
In case R = Q, R, or C, the smallest subfield is Q.
Z/pZ has no smaller subfields.
Q is the smallest subfield of Q(X) and of
R(X).
Clearly, Q is the smallest subfield of Q(X) and
of R(X). If p is a prime number,
Z/pZ is the smallest subfield of
Z/pZ[X]/(f)
if f is irreducible in Z/pZ[X].
The smallest subfield of Q+Qi is Q.