dimashenme / trivium

An English translation of a problem course ("Trivium") in undergraduate maths

This is an English translation of a course that was taught by Misha Verbitsky and Dmitry Kaledin at the Independent University of Moscow in the fall of 2004 (also known under the name “Trivium”). The course was targeted at first-year students, and the prerequisites include only elementary high-school mathematics. It covers a broad range of material that the authors believe is fundamental for one's mathematical education, and culminates in algebraic Galois theory and Galois coverings theory. It builds upon simple and straightforward definitions and lots of exercises, so the only serious requirement from someone taking the course is to be patient and steady.

The students who took this course in Moscow were presented with the following set of rules: for each chapter they had to either solve all the problems without an asterisk or solve all the problems marked with an asterisk (in case they thought that the problems without an asterisk were too easy or boring). Problems marked with “!” are vital for the development of the course and must have been solved by everyone. If one had solved k problems with double asterisk than they had a right to skip 2k problems with a simple asterisk.

The repository contains the original Russian LaTeX source of the course as well as the English translation.

Compiled PDFs are available at http://shenme.de/listki/

Table of contents

Algebra

1. Groups, rings and fields.
2. Divisibility in rings and Euclid's algorithm.
3. Vector spaces and linear maps.
4. Algebraic numbers.
5. Algebras over a field.
6. Grassmann algebra and determinant.
7. Matrices and determinants.
8. Linear algebra: characteristic polynomial.
9. Artinian rings and idempotents.
10. Normal groups and representations.
11. Galois theory.
12. Semisimple and nilpotent operators.

Geometry

1. Real numbers.
2. Real numbers, part 2.
3. Metric spaces and norm.
4. Topology of metric spaces.
5. Set-theoretic topology.
6. Set-theoretic topology: product of spaces.
7. Set-theoretic topology: compactness.
8. Pointwise and uniform convergence.
9. Connectedness.
10. The fundamental group and the loop space.
11. Galois coverings.
12. Fundamental group and homotopies.

-- translation by Dima Sustretov ([e-mail]), 2016