If you're taking this course, please enroll on Jogustine. This course is the follow-up of the course Algebra 1 given the preceding semester by Manuel Blickle.

**Practical information**

**Teacher:** Ariyan Javanpeykar

**Exercise session:** Robert Wilms

**Course: **Tuesday 16h-18h and Wednesday 14h-16h in Room 04-426.

**Ubungsgruppe: **Wednesday 12h-14h in Room 04-512.

**Prerequisites: **Algebra 1

**Course material and description **

In this course, we continue our study of commutative rings. We will follow Atiyah-Macdonald's book for a large part. We will also use Looijenga's notes.

Here are some typed up notes for Vorlesung 7 and 8 by Robert Wilms: Nullstellensatz.

Here are some typed up notes for Vorlesung 23-26 by Robert Wilms: Krull

**Homework**

We recommend you write your solutions in LateX. This will increase the quality of your homework, and it's a great way to practice writing mathematics with LateX. To use LateX, download MikTeX and an editor (like TeXmaker or TeXworks). A good tutorial for using LateX can be found under __this link__.

- April 28th. Blatt 1
- May 5th. Blatt 2
- May 15th. Blatt 3
- May 19th.Blatt 4
- May 26th. Blatt 5
- June 2nd. Blatt 6
- June 9th. Blatt 7
- June 16th. Blatt 8
- June 23rd.Blatt 9
- June 30th. Blatt 10
- July 7th. Blatt 11
- July 14th. Blatt 12

**ORAL EXAMS: Tuesday, July 25th 2017. (You need at least 72 points to get the Prufungszulassung.)**

Check your points here; Punkte_bis_Blatt11.

**Schedule**

- Tuesday April 18th. Rings, division algebra's, Hamilton quaternions, polynomial rings, matrixrings, endomorphism rings of abelian groups, zero divisors, integral domains, homomorphism theorem, isomorphism theorem, Chinese remainder theorem.
- Wednesday April 19th. Proof of Chinese remainder theorem. Division with remainder. Principal ideal domains (PIDs). Cyclicity of finite subgroups of R* (after Gauss). The unit group of a finite field is cyclic. The polynomial ring K[x] over a field K is a PID. Irreducible elements. Nonzero elements in a PID factor.
- Tuesday April 25th. Prime elements. Irreducible elements in a PID are prime. Unique factorization in PIDs (hence K[x]). Prime ideals and unique factorization of prime ideals in PIDs. Example: Z[i] is a PID. Decomposition of prime numbers in Z[i].
- Wednesday April 26th. Solving the Diophantine equation y^2=x^3+1. Euclidean rings. Examples: Z[rho] and Z[sqrt{-2}] and Z[sqrt{-5}]. Factorizing polynomials. Unique factorization domains.
- Tuesday May 2nd. Algebraically closed fields. Maximal ideals and Zorn's lemma. The Zariski topology on \mathbb{A}^n; from ideals to algebraic sets. Regular functions on \mathbb{A}^n. The coordinate ring of a closed subset of \mathbb{A}^n. Hilbert's Nullstellensatz (proof later). Consequence of HNS: correspondence between closed subsets and radical ideals, points and maximal ideals.
- Wednesday May 3rd.
**No Lecture.** - Tuesday May 9th. (Robert Wilms) Section 2 in Looijenga's notes.
- Wednesday May 10th. (Robert Wilms) Section 3 in Looijenga's notes.
- Tuesday May 16th. Irreducibility, irreducible components of noetherian spaces, irreducible components and prime ideals.
- Wednesday May 17th. Cayley-Hamilton. Krull dimension of a ring. Dimension of a noetherian irreducible topological space. Main theorem: dim A^n = n. (Proof later.) Definition of projective space.
- Tuesday May 23rd. Projective Hilbert's Nullstellensatz. Examples of projective varieties.
- Wednesday May 24th. Morphisms of quasi-projective varieties. For affine varieties: Coordinate ring = ring of regular functions. Function field of a variety. Local ring of a variety a point.
- Tuesday May 30th. Equivalence of categories: Affine varieties and affine k-algebra's. D(f) is isomorphic to an affine variety. Examples. Frobenius.
- Wednesday May 31st. The cuspidal curve y^2=x^3. Products of quasi-affine varieties.
- Tuesday June 6th.Statement of main theorem of elimination theory. projective varieties are complete. Properties of complete varieties (e.g, O(X) =k). If f:X->Y is surjective, and X and Y are projective, then dim X >= dim Y.
- Wednesday June 7th. dim P^n = n by using projections from a point (and assuming main theorem of elimination theory).
- Tuesday June 13th. Proof of main theorem of elimination theory. Veronese embedding.
- Wednesday June 14th.
**No Lecture** - Tuesday June 20th. Dimension of A^n =n.
- Wednesday June 21st. Dimension and families of varieties. Local noetherian rings and tangent spaces.
- Tuesday June 27th. Cotangent space. Examples. Basic properties of local regular rings.
- Wednesday June 28th.
- Tuesday July 4th. (Robert Wilms)
- Wednesday July 5th. (Robert Wilms)
- Tuesday July 11th. (Robert Wilms)
- Wednesday July 12th. (Robert Wilms)