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.