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Computation in Algebra and Number Theory
Last Updated: 2026-02-05 16:06:49
Abstract
This course will cover explicit, computational methods in a selection of areas of algebra and number theory. The lectures will survey the ideas needed in order to make the relevant objects explicit enough to represent on a computer, and a selection of the important algorithms; the exercise classes will give a hands-on introduction to some of the available software.
Content
Course content (approximate): - Commutative algebra: ideals in polynomial rings; Groebner bases and Buchberger's algorithm; elimination theory - Algebraic geometry: computing with varieties in affine/projective space; elliptic curves (group structure, hints at applications to cryptography) - Polynomials over integers and finite fields: Galois groups, Hensel lifting, Zassenhaus factorization - Lattices: short vectors, LLL reduction, applications - Algebraic number fields: integer rings, ideals, class groups; number-field sieve - Group theory: presentations of groups, coset enumeration; SL(2, Z) and its subgroups. Representations and characters of finite groups, Burnside's algorithm.
Resources
Literature
Cox, Little + O'Shea "Ideals, varieties and algorithms" Stewart + Tall "Algebraic number theory and Fermat's last theorem" Cohen "A course in computational algebraic number theory" Rotman "Introduction to the theory of groups"
Learning Materials (Links)
- Main link
- Course web page
General Information
- Language
- English
- Levels
- BSC , MSC
Examination
- Type
- graded semester performance
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture with exercise | Computation in Algebra and Number Theory |
|
3 h weekly |
Offered In
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Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
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