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Durham University

Department of Mathematical Sciences

MATH3041 Galois Theory III

The origin of Galois theory lies in attempts to find a formula expressing the roots of a polynomial equation in terms of its coefficients. Evarist Galois was the first mathematician to investigate successfully whether the roots of a given equation could in fact be so expressed by a formula using only addition, subtraction, multiplication, division, and extraction of n-th roots. He solved this problem by reducing it to an equivalent question in group theory which could be answered in a number of interesting cases. In particular he proved that, whereas all equations of degree 2, 3 or 4 could be solved in this way, the general equation of degree 5 or more could not. Galois theory involves the study of general extensions of fields and a certain amount of group theory. Its basic idea is to study the group of all automorphisms of a field extension. The theory has applications not only to the solution of equations but also to geometrical constructions and to number theory.

Outline of Course

Aim: To introduce the way in which the Galois group acts on the field extension generated by the roots of a polynomial, and to apply this to some classical ruler-and-compass problems as well as elucidating the structure of the field extension.

Term 1

  • Introduction: Solving of algebraic equations of degree 3 and 4. Methods of Galois theory.
  • Background: Rings, ideals, a ring of polynomials, fields, prime subfields, factorisation of polynomials, tests for irreducibility.
  • Field extensions: Algebraic and transcendental extensions, splitting field for a polynomial, normality, separability.
  • Fundamental theorem of Galois theory: Statement of principal results, simplest properties and examples.

Term 2

  • Galois extensions with simplest Galois groups: Cyclotomic extensions, cyclic extensions and Kummer theory, radical extensions and solvability of polynomial equations in radicals.
  • General polynomial equations: Symmetric functions, general polynomial equation and its Galois group, solution of general cubic and quartic, Galois groups of polynomials of degrees 3 and 4, non-solvability in radicals of equations of degree ≥5.
  • Finite fields: Classification and Galois properties, Artin-Schreier theory, construction of irreducible polynomials with coefficients in finite fields.

Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.

Reading List

Please see the Library Catalogue for the MATH3041 reading list.

Examination Information

For information about use of calculators and dictionaries in exams please see the Examination Information page in the Degree Programme Handbook.