I am reading Dummit and Foote Section 9.6 Polynomials in Several Variables Over a Field and Grobner Bases.
I have a second problem (see previous post for first problem) understanding a step in the proof of Proposition 24, Page 322 of D&F
Proposition 24 reads as follows:
Proposition 24. Fix a monomial ordering on and let I be a non-zero ideal in R
(1) If are any elements of I such that
then is a Grobner Basis for I
(2) The ideal I has a Grobner Basis
The proof of Proposition 24 begins as follows:
Proof: Suppose with .
We need to see that generate the ideal I.
If use general polynomial division to write where no non-zero term in the remainder r is divisible by any
Since , also , which means LT9r) is in LT(I).
But then LT(r) would be divisible by one of , which is a contradiction unless r= 0 ... ... etc etc
... ... ...
My problem is with the last (bold) statement - as follows:
We have that
Now since , then we have that r is a finite sum of the form
... ... ... (*)
But surely (*) is NOT guaranteed to be divisible by for some i
Can someone please clarify this issue?