Homework 5, March 14

Math 402 – Spring Semester, 1990 Second Course in Abstract Algebra, II Prof. Daniel Grayson

2:00 PM MWF, Section F1, 243 Altgeld Hall

These problems are due Wednesday, March 28, after break.

Problem 1: Compute a Gröbner basis for the equations

a^{3} = bc
b^{3} = ac
c^{3} = ab

When we discussed this in class, we were thinking of an ordering for monomials where higher degree monomials were always bigger, but now I want you to use the lexicographic ordering on monomials. Using this Gröbner basis, produce a set of monomials, all of which are minimal elements of the Gröbner graph, which serve an F -vector space basis for the quotient ring.

Problem 2: Compute a Gröbner basis for the equations

x^{3} + y^{3} = 11
x^{2} + y^{2} = 1

Can you say anything about the number of solutions of these equations over the complex numbers?

Problem 3: Let F be a field, let R = F [T_{1}, …, T_{n}], let I = Rf_{1} + … Rf_{t} be an ideal of R, and let F be the family of replacement rules corresponding to the generators f_{1}, …, f_{t}. Show that the graph Γ(F) is confluent provided the confluency property holds for any pair of arrows originating in a monomial.

The following exam question is to be done independently.

Exam Problem 2: Let F be a field, and let R = F [T_{1}, …, T_{n}]. We consider a total ordering of the monomials in R such that each variable T_{i} is greater than 1, and if p and q are monomials satisfying p ≤ q, then pT_{i} ≤ qT_{i}. Show that the monomials are well-ordered by this ordering.