Fachgebiet Algorithmische AlgebraAlgebra II

# Algebra II: Commutative Algebra

• 4h Course, 10 ECTS
• Eligible as BMS Basic Course in area 2

This course is the continuation of the Algebra I course given by Dr. Dirk Kussin at TU Berlin in WS 20/21.
The course Algebra I explained the basic notions of algebra: groups, rings, fields, factor structures, and provided a fairly detailed treatment of algebraic field extensions, culminating in the beautiful Galois theory.

## Schedule and Organization

Please note that both lectures start at 12:00 sharp.
The course starts on April 13.

Vorlesungszeiten
Type
Day
Zeitraum
Format
Lehrperson
Lecture
Tuesday
1200 - 1330
online
Prof. Dr. Peter Bürgisser
Lecture
Thursday
1200 - 1330
online
Prof. Dr. Peter Bürgisser
Exercise
Wednesday
1000 - 1200
online
Dr. Marco Ramponi

Due to the current situation, the course will be given online via Zoom. Complementary material accompanying the lecture will be provided on ISIS.

People interested to participate in the course are asked to send an email to Peter Bürgisser (pbuerg at math.tu-berlin.de) with cc to Marco Ramponi (ramponi at math.tu-berlin.de).

## Description

The main goal of Algebra II is to provide an introduction to commutative algebra. This is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Important examples of commutative rings are rings of algebraic integers (which includes the ring of integers) and polynomial rings over fields and their factor rings. Both algebraic geometry and algebraic number theory build on commutative algebra. In fact, commutative algebra provides the tools for local studies in algebraic geometry, much like multivariate calculus is the main tool for local studies in differential geometry.

Probably the best entry to the subject is the following short, concise, and clearly written textbook:
Atiyah, M.F.; Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co, 1969 ix+128 pp.

The current plan is to follow this book quite closely. We plan to complement this with some additional material concerning algorithms (Gröbner bases).

## Literature

• Atiyah, M.F.; Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co, 1969 ix+128 pp.
• Lang S., Algebra, 3rd edition, Springer, 2002
• Algebra I+II, Vorlesungsskript von Prof. Hess an TU Berlin