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