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MX4082: GALOIS THEORY (2017-2018)

Last modified: 23 Aug 2017 15:37


Course Overview

Galois theory is based around a simple but ingenious idea: that we can study field extensions by instead studying the structure of certain groups associated to them. This idea can be employed to solve some problems which confounded mathematicians for centuries, including the impossibility of trisecting an angle with ruler and compass alone, and the insolubility of the general quintic equation.



Course Details

Study Type Undergraduate Level 4
Session First Sub Session Credit Points 15 credits (7.5 ECTS credits)
Campus None. Sustained Study No
Co-ordinators
  • Dr Jean-Baptiste Gramain

Qualification Prerequisites

  • Programme Level 4

What courses & programmes must have been taken before this course?

What other courses must be taken with this course?

None.

What courses cannot be taken with this course?

None.

Are there a limited number of places available?

No

Course Description

  • Field Theory, Field Extensions.
  • Constructible Numbers.
  • The Galois Group of a Field Extension.
  • Cyclotomic Fields.
  • Splitting Fields of Polynomials.
  • Normal Extensions, Separable Extensions.
  • Simple Fields Extensions.
  • Counting Field Homomorphisms.
  • Galois Extensions.
  • The Galois Correspondence.
  • Cyclic Galois Groups.
  • Radical Extensions and Solvable Galois Groups.
  • The Galois Group of a Polynomial. Applications.

 

Syllabus

  • Constructible numbers.
  • The impossibility of trisecting an angle and squaring the circle.
  • Normal and separable field extensions. Galois extensions. Galois groups.
  • The Galois correspondence.
  • The Galois group of a polynomial.
  • The use of Galois theory in the solution of polynomial equations.
  • The impossibility of solving general polynomial equations of degree five or greater by radicals.

Further Information & Notes

Course Aims
Galois theory, so named after the French mathematician Évariste Galois (1811–1832), combines field
theory and group theory to one of the highlights in algebra, yielding insight into a great diversity of mathematical problems. At the origin of this theory are questions related to the solvability of polynomial equations by radicals. We will in particular show the classical result that polynomial equations of degree at least five are not always solvable by radicals. In the process we will also obtain a description of which constructions are possible with a straight edge and a compass - proving in particular that it is impossible, in general, to trisect a given angle, to square the circle, to double the cube, or to construct a regular heptagon. The topics covered include
- Constructible numbers.
- The impossibility of trisecting an angle and squaring the circle.
- Normal and separable field extensions.
- Galois extensions.
- Galois groups.
- The Galois correspondence.
- The Galois group of a polynomial.
- The use of Galois theory in the solution of polynomial equations.
- The impossibility of solving general polynomial equations of degree five or greater by radicals.

 

Learning Objectives
By the end of the course the student should:
- be able to determine whether a given complex number is constructible;
- be able to calculate Galois groups of certain field extensions;
- be able to calculate Galois groups of certain polynomials;
- be able to decide whether a given polynomial equation is solvable by radicals;
- understand the Galois correspondence and more generally, theoretical aspects of the interplay between field theory and group theory applied to the structure theory of fields and solvability of polynomial equations.

Degree Programmes for which this Course is Prescribed

  • BSc Computing Science-Mathematics
  • BSc Mathematics
  • BSc Mathematics with French
  • BSc Mathematics with Gaelic
  • MA Business Management - Mathematics
  • MA Computing and Mathematics
  • MA French-Mathematics (Mode B)
  • MA Hispanic Studies-Mathematics
  • Mathematics Joint
  • Mathematics Major

Contact Teaching Time

56 hours

This is the total time spent in lectures, tutorials and other class teaching.

Teaching Breakdown


Assessment

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%). Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).

Only the marks obtained on first sitting can be used for Honours classification.

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinators for feedback on the final examination.

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