Lecturer: Harry Schmidt

Term(s): Term 1

Status for Mathematics students: List A

Commitment: 30 lectures plus assessment sheets

Assessment: 85% 3 hour examination, 15% best 3 out of 4 assessed worksheets

Formal registration prerequisites: None

Assumed knowledge:

• MA106 Linear Algebra: Linear independences, bases of vector spaces, dimension, linear maps, rank-nullity formula
• MA138 Sets and Numbers: Factorisation of polynomials, long division and the Euclidean algorithm
• MA249 Algebra II: Groups and Rings: Fields, rings and ideals, quotient rings and the first isomorphism theorem - especially polynomial rings and their quotient rings by ideals. Groups and homomorphisms, normal subgroups and quotient groups and the isomorphism theorems.

Useful background: Whilst we do use the technical machinery described as assumed knowledge, at some level this module is extremely hands on, and you will benefit by practising the Euclidean algorithm for polynomials, working with permutations (in permutation groups), and manipulating complex numbers (inverses, solving quadratic polynomials, and especially the roots of unity).

Synergies:

• MA3A6 Algebraic Number Theory:

Although each subject is more general in differing respects, the fundamental objects of study in each module include fields that contain the rational numbers and are finite dimensional as a vector space over them - the set of all complex numbers you can write using rationals and the square root of 2 is an example. In Galois theory we study the symmetries of such fields, while in Algebraic Number Theory the focus is on number-theoretic questions, such as questions about factorisation.

• MA3K4 Introduction to Group Theory:

Galois Theory uses groups of permutations and their subgroups as fundamental objects that capture the symmetry of field extensions and of solutions of polynomials. Any familiarity with permutations and groups is good, and in particular soluble groups appear in both modules: in Galois Theory they capture the symmetries that arise when you repeatedly extract square roots, cube roots and higher.

Leads to: The following modules have this module listed as assumed knowledge or useful background:

• MA4J8 Commutative Algebra II
• MA453 Lie Algebras
• MA4M3 Local Fields
• MA4L7 Algebraic Curves
• MA4H8 Ring Theory

Content: Galois theory is the study of solutions of polynomial equations. You know how to solve the quadratic equation $ ax^2+bx+c=0 $ by completing the square, or by that formula involving plus or minus the square root of the discriminant $ b^2-4ac $ . The cubic and quartic equations were solved ``by radicals'' in Renaissance Italy. In contrast, Ruffini, Abel and Galois discovered around 1800 that there is no such solution of the general quintic. Although the problem originates in explicit manipulations of polynomials, the modern treatment is in terms of field extensions and groups of ``symmetries'' of fields. For example, a general quintic polynomial over $Q$ has five roots $ \alpha_1.\dots.\alpha_5 $ , and the corresponding symmetry group is the permutation group $ S_5 $ on these.

Aims: The course will discuss the problem of solutions of polynomial equations both in explicit terms and in terms of abstract algebraic structures. The course demonstrates the tools of abstract algebra (linear algebra, group theory, rings and ideals) as applied to a meaningful problem.

Objectives: By the end of the module the student should understand:

• Solution by radicals of cubic equations and (briefly) of quartic equations
• The characteristic of a field and its prime subfield. Field extensions as vector spaces
• Factorisation and ideal theory in the polynomial ring k[x]; the structure of a simple field extension
• The impossibility of trisecting an angle with straight-edge and compass
• The existence and uniqueness of splitting fields
• Groups of field automorphisms; the Galois group and the Galois correspondence
• Radical field extensions; soluble groups and solubility by radicals of equations
• The structure and construction of finite fields

Books: DA Cox, Galois Theory, Wiley (second edition 2012).

IN Stewart, Galois Theory, CRC Press (e.g. fourth edition 2015, or any other edition).

DJH Garling, A Course in Galois Theory, CUP (there must be later editions than my 1986 one).

Additional Resources