Commutative algebra 32 Zariski's lemma

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.

We state and prove Zariski's lemma: Any field that is a finitely generated algebra over a field is a finitely generated module.
As applications we prove the nullstellensatz for all algebraically closed fields. We also show that for homomorphisms of finitely generated algebras over a field, the inverse image of a maximal ideal is maximal.

Typo on sheet 4: "finite k[x1....xm] module" should be "finite k[x1...xm, 1/b{m+1}...] module"

Reading: Section 4.5
Exercises: 32