Suggestions about exercise 10 in Sec 2.6

Updated October 29, 2008

1. First, a basic definition: saying that two monomials are relatively prime means that they have no common factor.
For instance, in 3 variables, xy and xz are not relatively prime because y is a common factor. On the other hand,
xy and z² are relatively prime.

2. Now a question: if two monomials are relatively prime, what is their LCM? Similar terminology is used for integers.
For instance, 6 and 25 are relatively prime: what is their LCM? A very similar answer is valid for relatively prime monomials.

3. The equation that you're asked to prove in part a certainly is true, but something slightly different may be more useful in part b.
First, note that fg is both added and subtracted in the expression on the right side. Therefore, that equation is equivalent to
the following equation:

S(f,g) = LT(g)f - LT(f)g
And remarkably, this equation is both more useful in part b {in my opinion, anyway} and also easier to prove.
{Just make appropriate substitutions from what is given, and the answer to the question asked above,
and then do a bit of algebraic calculation.}

Comments and questions to: roberts@math.umn.edu

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