This course is the continuation of the Algebra I course given by Dr.~Dirk Kussin at TU Berlin in WS 20/21.
It is eligible as BMS Basic Course in area 2.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.
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. Online available from Fachbibliothek TU Berlin.
We shall follow this book quite closely, but plan to complement it with some additional material concerning algorithms (Gröbner bases).
Due to the current situation, the course will be given online via Zoom.
Lecture notes and complementary material will be provided on ISIS.
- Trainer/in: Peter Bürgisser