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This second part of the Level 4 honours algebra sequence centres on Galois Theory, which arose from the search for explicit formulae for roots of polynomial equations. The primary focus of the course is the Galois Correspondence Theorem, relating the structure of fields to the structure of groups, and its applications to polynomials.

17 x 1 hr lectures and 6 x 1 hr tutorials in a semester

Mandatory Entry Requirements

4H: Topics in Algebra - MATHS4111

Recommended Entry Requirements

Galois Theory (MATHS4014)

Assessment

90% final exam, 10% coursework.

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

This course focuses on work of Galois which led to a satisfactory framework for fully understanding the fact that the general polynomial equation of degree at least 5 could not always be solved `by radicals' (i.e., with formulae involving n-th roots, analogous to the well-known Quadratic Formula). Further, the so-called Galois Correspondence allows the structure of fields to be related to the structure of groups, a powerful idea with applications in many fields of mathematics. Topics to be covered in this course include field extensions, algebraic extensions, algebraic closure, separable and normal extensions, Galois extensions, finite fields, cyclotomic fields, symmetric functions, and solvability, as time permits

By the end of this course students will be able to:

a) state and apply criteria for determining whether a given polynomial is irreducible;

b) find the irreducible polynomial of an element over a field;

c) compute the degree of a field extension;

d) determine the group of automorphisms of an extension field over a base field;

e) describe and calculate values of the Frobenius automorphism;

f) state, prove, and apply the Isomorphism Extension Theorem;

g) calculate the index of a field extension, and prove it divides the degree;

h) decide whether a given polynomial splits in an extension field;

i) prove that the roots of a polynomial all have the same multiplicity in an algebraic closure; 

j) state and prove the Primitive Element Theorem;

k) state, prove, and apply the Galois Correspondence Theorem (aka the Fundamental or Main Theorem of Galois Theory).

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.