Algorithms for determining if an ideal in k[x1,...,xn] is maximal.

George Zhang, Simon Fraser

Let k be a field, and I be an ideal in the polynomial ring k[x1,...,xn].
Suppose we want to do computations in the quotient ring k[x1,...,xn]/I.
We will want to know if R is a field, and if not, whether R is an integral
domain.  A well known theorem in algebra says that

    (i) R is an integral domain <==> I is prime, and
   (ii) R is a field <==> I is maximal.

How then can we test if a given ideal I is prime or maximal?
We will prove the following theorem which will give us a constructive
test for maximality of I

  (iii) I is maximal <==> I is prime and I is finite dimensional.

We show how to use Groebner bases to test if I is finite dimensional
and then to test if I is prime.