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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.