I was asked: In the proof of the Hilbert Basis Theorem when defining $g_1$ shouldn't it say that $g_1=\sum(r_i X^{\deg(g)-n} f_{i,n})$ ? Otherwise we won't necessarily get the same leading coefficient. and this is correct. In the printed notes (page 27 line 11) it incorrectly has $f_{1,n}$ instead of $f_{i,n}$.