Remark

Implicit in Part 3 is the fact any two irreducible polynomials in Z/pZ[X] of the same degree, say f and g, lead to isomorphic finite fields. The theorem does not give any information on how to construct the isomorphism. A way to proceed is to look for a zero y of g in Z/pZ[X]/(f), and to construct the isomorphism as the map Z/pZ[X]/(g) -> Z/pZ[X]/(f) sending X + (g) to y.

Later, we shall see that, for every prime power, there exists a field of that order.