| Lecture | Date | Content |
|---|---|---|
| 1 | 24.02. | Aims of the course. Exact computation. Exact structures: \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{Z}/n\), polynomials, matrices, etc. Non-example of \(\mathbb{R}\). Interval arithmetic. Polynomial root isolation and \(\bar{\mathbb{Q}}\). |
| 2 | 03.03 | Matrix algebra over fields, echelon form, subspaces. Matrix algebra over \(\mathbb{Z}\): Hermite and Smith form. Applications to abelian groups. | 3 | 10.03 | Univariate polynomials. Factorisation in \(\mathbb{F}_p[x]\). Extensions of finite fields, Conway polynomials. | 4 | 17.03 | Univariate polynomials over \(\mathbb{Q}\) and \(\mathbb{Z}\). Mignotte’s bound, factorisation by Hensel lifting. | 5 | 24.03 | Multivariate polynomials. Ideals. Vocabulary of algebraic geometry; affine / projective varieties. Term orderings, Groebner bases. | 6 | 31.03 | Buchberger’s algorithm; some more advanced algorithms. Elimination theory. Radicals, primary decomposition, etc. | 7 | 07.04 | More on algebraic geometry. Invariant theory (if time). | 8 | 14.04 | Elliptic curves (generalities). Elliptic curves over finite fields, applications to cryptography. | 21.04 | Easter break | 9 | 28.04 | Number fields I: integer rings, discriminant, ideal factorisation. | 10 | 05.05 | Number fields II: class group, unit group, regulator. Hints at class field theory. | 11 | 12.05 | Group theory: presentations of groups, coset enumeration, applications to \(\operatorname{SL}(2, \mathbb{Z})\). | 12 | 19.05 | Finite groups: representations and characters, Burnside’s algorithm. | 26.05 | No lecture (Ascension Day) | 13 | 02.06 | Overflow slot |
Algebra II, Commutative Algebra; basic notions of computer programming (e.g. first-year bachelors "Computer Science" course). Prior knowledge of algebraic number fields and/or algebraic varieties will be useful, but not absolutely necessary.
The new exercises will be posted here.
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| Sheet 10 | Notes |
| Notes |
Last updated: 02.06.2022