Existence of reduced Groebner basis

Question

In my class notes there is a Theorem saying: Every non zero ideal of the ring $F[x_1,\dots,x_n]$ has a reduced Groebner Basis. Unfortunately there is no proof. Can someone give the proof or refer to somewhere where I can find it? Thanks in advance

Answer 1

By definition a Groebner basis $G$ is reduced when for every $g \in G$,

• $LC(g) = 1$, and
• $LM(g)$ divides no monomial of any element of $G \setminus \{g\}$.

Here $LC(g)$ denotes the leading coefficient of $g$ and $LM(g)$ denotes the leading monomial of $g$, and of course these notions (and the notion of a Groebner basis) depend on a chosen monomial ordering.

Given an arbitrary Groebner basis $G'=(f_1,\ldots,f_s)$ we construct a reduced Groebner basis as follows:

1. Remove all "redundant" $f_i$ with $LM(f_j) \mid LM(f_i)$ for some $j$.

2. Replace each $f_i$ by $f_i/LC(f_i)$, to normalize all the leading coefficients.

3. Replace each $f_i$ by a reduction of $f_i$ by the set of remaining $f_j$ with $j \neq i$.

The last step affects only the non-leading terms (due to step 1), and it achieves exactly the second requirement for being reduced. QED.