## Suggestions about the exercises in Sec 2.5

Updated October 22, 2008

1. In exercises 1, 2, and 7, the recommended method is as follows:

• Try find an element  g  of the given ideal, such that  LT(g)  is
not in the ideal generated by the leading terms of the given polynomials  g1, ..., gs.
The most effective strategy involves doing calculations in which leading terms cancel  --  for instance,
you can multiply two of the given generators by monomials (or constant multiples of monomials)
so that cancellation will happen when you subtract.
We'll see in the near future that this amounts to calculating the S-polynomial  S(gi, gj).

• Probably not needed for these problems, but mentioned just in case  .  .  .
If one of these calculations doesn't produce a "new" leading term,
then divide by  g1, ..., gs  and take the remainder.

2. In exercise 17(a), prove the inclusion ⟨ x2 - y, x2 -2 ⟩ ⊂ ⟨ x2 - y, y + x2 - 4 ⟩,
by doing a calculation to show that  x2 -2 ∈ ⟨ x2 - y, y + x2 - 4 ⟩. Then do another
ideal membership calculation to show that the opposite set inclusion also holds.