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public:teaching:galoisth-w10

Galois theory (WS 2010/11)

  • Spezialvorlesung “Algebra und Diskrete Mathematik”
  • Lecturer: David Sevilla (david.sevilla@oeaw.ac.at)
  • Frequency: 2 hours/week lecture + 1 hour/week exercises
  • Language: English

Overview

Galois Theory is a basic part of abstract algebra which connects field theory with group theory. It was originally conceived as a tool to study solutions of polynomial equations, but it has other applications, like in the study of algebraic varieties.

In this course we will take an elementary approach, starting from simple ideas about polynomial roots, in a similar way to how Galois and others conceived it. Later on, we will learn the modern approach via field extensions and, if time permits, will see briefly some of the classical applications: equations solvable by radicals, geometric constructions in the plane with compass and straightedge. Ideally we would work with many concrete examples and also learn the abstract algebra well, but time will be scarce, so we will try to come to a compromise on the explicit and abstract aspects of the contents as we progress.

Grading system

The final grade will be a combination of:

  • Average of written assignments (25%). Almost every week I will give out some exercises to work on. Not all of them will count for the grade, I will say which do in advance. Work in groups is allowed (even encouraged), but every student should submit his/her individual solution and be ready to defend it (it is not enough at all to copy someone else's solution). To account for busy study periods, sickness, etc. the worst-graded assignment will be ignored.
  • A written midterm exam (25%). It is intended as a short, low-stress exercise, with some concrete numerical questions and some theoretical ones (for these, short answers and rough ideas will be enough to get full marks).
  • A final exam (50%). Again, a combination of concrete calculations and theoretical questions, but more accuracy is required than in the midterm. For those who work on the assignments, the questions should not be a surprise. We may agree to make an oral exam instead of a written one (this will not be as difficult as it seems, believe me).
  • Participation in class will be taken into account positively, in those cases where the student is close to getting a better grade (from 5 to 4, or from a passing grade to a better one).

Resources

Notes will be available shortly after each lecture, but normally before as well, in KUSSS.

Bibliography:

  • Jörg Bewersdorff, Algebra für Einsteiger (accessible online from the JKU online catalog) or Galois theory for beginners
  • J. S. Milne, Field and Galois Theory, online notes at www.jmilne.org/math/
  • Joseph J. Rotman, Galois theory
  • Ian Stewart, Galois theory
  • Jean-Pierre Tignol, Galois’ theory of algebraic equations
  • Lisl Gaal, Classical Galois theory

Lectures schedule

The schedule can be found in KUSSS. A rough table of contents:

  • Classical Galois theory: solvability by radicals, resolvents, original definition of the Galois group.
  • Modern Galois theory: field extensions, some group theory, field automorphisms, the modern definition of the Galois group, the fundamental theorem.
/var/www/wiki/htdocs/data/pages/public/teaching/galoisth-w10.txt · Last modified: 2011/03/08 15:32 by sevillad