Many properties of the polynomial ring K[X] discussed in Chapter 4 for special fields K = Q, R, C or Z/pZ with p prime are in fact valid for arbitrary fields K. For instance,
hold. Proofs can be copied verbatim, so we shall not repeat them. An important consequence is that we can compute modulo a polynomial d and construct the residue class ring K[X]/(d).
This allows us to construct new fields.
The residue class a + (d) has an inverse in K[X]/(d) if and only if gcd(a,d) = 1.
Let d be an irreducible polynomial in K[X]. Then S = R[X]/(d) is a field: Every nonzero element in S has an inverse.
Let K be a field. In the sequal we need the following general result, which extends a previous lemma.
Let g 
K[X].
- If x K is a zero of
g, then X - x divides g.
- If g has degree n, then g has at most n zeros in K.