This second part of an 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

4H: Galois Theory (MATHS4105)

Assessment

90% Examination, 10% Coursework.

Reassessment

Resit opportunities available for MSc students

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 unique factorization domains, principal ideal domains, euclidean domains, field extensions, algebraic extensions, algebraic closure, separable and normal extensions, Galois extensions, finite fields, cyclotomic fields, symmetric functions, solvability, and separable closures of a field, 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) construct the separable closure of a field in a totally inseparable field extension

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

m) prove that every principal ideal domain is a unique factorization domain;

n) perform arithmetic in Euclidean domains.

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