| # | Lecture |
|---|
| 1 | Prelude; rings and ideals |
| 2 | Operations on ideals; maximal and prime ideals |
| 3 | Radicals, local rings and modules; first examples |
| 4 | Polynomial and power series rings; the Hilbert basis theorem |
| 5 | Affine varieties; monomial orderings |
| 6 | Division with remainder; Gröbner bases |
| 7 | Buchberger's criterion and algorithm |
| 8 | The Hilbert function and the Hilbert polynomial |
| 9 | Dimension of a variety; towards the Nullstellensatz |
| 10 | Integral extensions |
| 11 | Noether normalisation |
| 12 | The weak Nullstellensatz |
| 13 | The strong Nullstellensatz |
| 14 | Irreducible decomposition; the Zariski topology |
| 15 | Regular maps; the theorem of Ax |
| # | Lecture |
|---|
| 16 | The spectrum of a ring |
| 17 | Krull dimension; Artinian rings |
| 18 | Localisation |
| 19 | Local rings: Nakayama's lemma and Krull's height theorem |
| 20 | Krull's principal ideal theorem |
| 21 | Proof of the height theorem; dimension of polynomial rings |
| 22 | The Cohen–Seidenberg theorems |
| 23 | Valuation rings |
| 24 | Valuations and discrete valuation rings |
| 25 | Artin–Rees, Krull's intersection theorem and the main theorem on DVRs |
| 26 | Dedekind domains |
| 27 | Completion of a discrete valuation ring |
| 28 | p-adic numbers and Hensel's lemma |
| 29 | I-adic completion |
| 30 | What we left out |
