Up to now we have encountered the following finite fields (p a prime): Z/pZ and Z/pZ[X]/(d) with d an irreducible polynomial. Here we determine the cardinality of such fields.
Let p be a prime number and n a positive integer.
If d is an irreducible polynomial in Z/pZ[X] of degree n, then Z/pZ[X]/(d) is a field with exactly pn elements.
The result above does not say much about the existence of
finite fields. The following statement will be proved in Chapter 7.
There exists an
irreducible polynomial f of degree n in
Z/pZ[X]. The residue class ring
Z/pZ[X]/(f) is a finite field having
pn elements.
Any finite field can be constructed in this way.