B2.2 Commutative Algebra (2019-2020)
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Rings and Modules is essential. Representation Theory and Galois Theory are recommended.
16 lectures
Assessment type:
- Written Examination
Amongst the most familiar objects in mathematics are the ring of integers and the polynomial rings over fields. These play a fundamental role in number theory and in algebraic geometry, respectively. The course explores the basic properties of such rings.
Modules, ideals, prime ideals, maximal ideals.
Noetherian rings; Hilbert basis theorem. Minimal primes.
Localization.
Polynomial rings and algebraic sets. Weak Nullstellensatz.
Nilradical and Jacobson radical; strong Nullstellensatz.
Integral extensions. Prime ideals in integral extensions.
Noether Normalization Lemma.
Krull dimension; dimension of an affine algebra.
Noetherian rings of small dimension, Dedekind domains.
- M. F. Atiyah and I. G. MacDonald: Introduction to Commutative Algebra, (Addison-Wesley, 1969).
- D. Eisenbud: Commutative Algebra with a view towards Algebraic Geometry, (Springer GTM, 1995).
Please note that e-book versions of many books in the reading lists can be found on SOLO and ORLO.