B2.2 Commutative Algebra (2024-25)
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- Lecturer: Profile: Dawid Kielak
Course information
General prerequisites:
Rings and Modules is essential. Galois Theory is strongly recommended.
Course term: Hilary
Course lecture information: 16 lectures
Course weight: 1
Course level: H
Assessment type: Written Examination
Course overview:
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.
Course synopsis:
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.
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.
Section outline
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Opened: Saturday, 11 January 2025, 12:00 AMDue: Monday, 3 February 2025, 9:00 AM
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Opened: Monday, 3 February 2025, 5:37 PMDue: Monday, 17 February 2025, 9:00 AM
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Opened: Friday, 28 February 2025, 3:07 PMDue: Monday, 3 March 2025, 9:00 AM
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Opened: Sunday, 23 March 2025, 7:05 PMDue: Monday, 28 April 2025, 9:00 AM
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Registration start: Monday, 13 January 2025, 12:00 PMRegistration end: Friday, 14 February 2025, 12:00 PM
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Class Tutor's Comments Assignment
Class tutors will use this activity to provide overall feedback to students at the end of the course.
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