We proved Buchberger's criterion, that if the S-polynomials reduce to 0 then anything in I does.
Three key insights:
1) When the reduction algorithm works, it doesn't just say "yes, p is in the ideal"; rather, by keeping track of the reductions it gives a specific way to write p = sum_i r_i g_i (where the g_i are the generators).
2) In that expression p = sum_i r_i g_i, each init(r_i g_i) is as cheap or cheaper than init(p).
3) The initial term of an S-polynomial S(g_i,g_j) is strictly cheaper than the LCM of the initial monomials.
With those in mind, it should be easier to read this proof, which is the one we followed in class. (Warning: he writes "F" sometimes for a general field, or even "Fq", rather than using the complex numbers, but it doesn't make any difference in this proof.)
