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Grobner Bases

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
I am reading Dummit and Foote Section 9.6 Polynomials in Several Variables Over a Field and Grobner Bases.

I have a 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 [TEX] R= F[x_1, ... , x_n] [/TEX] and let I be a non-zero ideal in R

(1) If [TEX] g_1, ... , g_m [/TEX] are any elements of I such that [TEX] LT(I) = (LT(g_1), ... ... LT(g_m) )[/TEX]

then [TEX] \{ g_1, ... , g_m \} [/TEX] is a Grobner Basis for I

(2) The ideal I has a Grobner Basis


The proof of Proposition 24 begins as follows:


Proof: Suppose [TEX] g_1, ... , g_m \in I [/TEX] with [TEX] LT(I) = (LT(g_1), ... ... LT(g_m) )[/TEX] .

We need to see that [TEX] g_1, ... , g_m [/TEX] generate the ideal I.

If [TEX] f \in I [/TEX] use general polynomial division to write [TEX] f = \sum q_i g_i + r [/TEX] where no non-zero term in the remainder r is divisible by any [TEX] LT(g_i) [/TEX]

Since [TEX] f \in I [/TEX], also [TEX] r \in I [/TEX] ... ... etc etc



My question: How do we know [TEX] f \in I \Longrightarrow r \in I [/TEX]?

Would appreciate some help

Peter

[This has also been posted on MHF]