University of Trieste - AY 2025/26

Advanced Algebra (Istituzioni di Algebra Superiore)

Course of Study: Mathematics

Alessandro Logar

Here it is possible to find some material concerning the course in Advanced Algebra. In particular you can find: the program of the course (of this academic year and of the past years), several links to books which are good references for the program of the course, a short summary of each lecture, lectures and video recordings of the course given in the past years, further material, like some written lectures and some examples.
All the lectures are recorded and can be found on the ``Teams'' platform. The video recording will be deleted after one year.

Academic year 2025-26.

Class schedule.

Monday, 3PM-5PM room 3B H2bis
Thursday, 3PM-5PM room 5A H2bis

First lecture: Monday 22nd 2025.

Summary of the lectures

Here you can find a short lesson diary.

Academic year 2020-21: summary of the lectures and video recording: See here (expired)

Academic year 2021-22: summary of the lectures and video recording: See here (expired)

Academic year 2022-23: summary of the lectures and video recording: See here (expired)

Academic year 2023-24: summary of the lectures: See here

Academic year 2024-25: summary of the lectures: See here

Useful material

Course summary

Useful links

Link to by Joseph J. Rotman, here you can find most of the contents of the course.
Link to Algebraic Number Theory and FLT by I. Stewart e D. Tall.
Link to Algebraic Number Theory, a computational approach by W. Stein.
Link to Introduction to Commutative Algebra by M. F. Atyah e I.G. Macdonald.
Another link to Introduction to Commutative Algebra by M. F. Atyah e I.G. Macdonald.
Algebra, by Mac Lane, Birkhoff.
Link to Algebraic Number Theory by Mattew Backer.

Some lectures (some in Italian, some in English)

Proof of the fact that all the bases of a free module have the same cardinality.

Examples (some in Italian, some in English)

Modules over a PID/Moduli su un PID

An example of a module over a PID
Un primo esempio con i gruppi abeliani.
Decomposizione di un modulo con gli invarianti fondamentali
Decomposizione di una matrice
Representation of an endomorphism as the direct sum of the companion matrices of the fundamental invariants.

Esempio riassuntivo (invarianti fondamentali, divisori elementari, forma canonica razionale di una matrice, forma canonica di Jordan).

Primary Decomposition/Decomposizione primaria

Decomposizione primaria
Ten points on a cubic curve (how to use primary decomposition to solve a geometric problem).
The three altitudes of a triangle meet in a common point.

Anelli artiniani

Alcuni esempi di anelli artiniani.