Let R be a field (think of Q, R, C or Z/pZ with p a prime) and d, a two polynomials in R[X]. Assume d is nonconstant.

The extended Euclidean algorithm gives a method for finding the inverse of a + (d) in R[X]/(d).

Theorem

The residue class a + (d) has an inverse in R[X]/(d) if and only if gcd(a,d) = 1.


This theorem allows us to construct new fields.

Corollary

Let d an irreducible polynomial in R[X]. Then S = R[X]/(d) is a field, i.e., every nonzero element in S has an inverse.


If R = Z/pZ, then S is finite. In the next section we will take a closer look at such residue class rings.