Commutative algebra 6 (Proof of Hilbert's basis theorem)

This lecture is part of an online course on commutative algebra, following the book
"Commutative algebra with a view toward algebraic geometry" by David Eisenbud.

In this lecture we prove Hilbert's basis theorem that ideals of polynomial rings are finitely generated. We first do this by proving that the ring of polynomials over a Noetherian ring is Noetherian. Then we adapt the proof to show the same result for power series rings. Finally we give Gordan's proof using
the result that any set of monomials has only a finite number of minimal elements.

Reading: Section 1,4
Exercises: 15.15 a

[efi]

When you talked about how the proof is not entirely constructive and a basis can be hard to find, I started to understand why my professor decided to squeeze in Gröbner Bases and the Buchberger Algorithm into his lecture. But sadly, for me personally, it made the lecture overloaded. Still cool stuff, though.

[TheAlpacaMan]

What part of the alternative proof depends on the original ring being Noetherian?

[FractalMannequin]

Minimal monomials in Dickson's lemma is basically existence of Pareto optimals (no wonder, the partial order is defined the same way).

[bettydandash]

Please could you give me an example about Dicksons lemma, and the solution of it
For example if I have an ideal I generated by <x^4 y^2, x^3 y^4, x^2 y^5>, how could I apply Dicksons lemma here?

[elhcs.]

can u explain the last step in proving that R[[x]]is noetherian ? how did u get rid of the infinit sum problem ?

[mouchenqi]

Paul Gordan, Poor Gordan: for the missing credits.