MT5876 Galois Theory

Academic year

2024 to 2025 Semester 2

Further information on which modules are specific to your programme.

Key module information

SCOTCAT credits

15

The Scottish Credit Accumulation and Transfer (SCOTCAT) system allows credits gained in Scotland to be transferred between institutions. The number of credits associated with a module gives an indication of the amount of learning effort required by the learner. European Credit Transfer System (ECTS) credits are half the value of SCOTCAT credits.

SCQF level

SCQF level 11

The Scottish Credit and Qualifications Framework (SCQF) provides an indication of the complexity of award qualifications and associated learning and operates on an ascending numeric scale from Levels 1-12 with SCQF Level 10 equating to a Scottish undergraduate Honours degree.

Availability restrictions

Module runs in alternating even years, 2020/21, 2022/23, 2024/25, etc

Planned timetable

11am Monday (odd weeks), Wednesday, Friday

This information is given as indicative. Timetable may change at short notice depending on room availability.

Module coordinator

Dr Y Len

Dr Y Len
This information is given as indicative. Staff involved in a module may change at short notice depending on availability and circumstances.

Module Staff

Dr Thibault Poiret

This information is given as indicative. Staff involved in a module may change at short notice depending on availability and circumstances.

Module description

Galois Theory is one of the most beautiful areas of mathematics, establishing a remarkable connection between the theory of polynomial equations and their roots, and group theory. The subject brings together ideas from the theory of groups and fields in a powerful way, culminating in the Fundamental Theorem of Galois Theory and Galois's Great Theorem. A consequence will be the demonstration that there is no general formula for the solution of quintic equations. There are many additional applications of this theory, for example, the demonstration that certain ruler and compass constructions are impossible.

Relationship to other modules

Pre-requisites

BEFORE TAKING THIS MODULE YOU MUST PASS MT3505

Anti-requisites

YOU CANNOT TAKE THIS MODULE IF YOU TAKE MT5836

Assessment pattern

2-hour written examination = 100%

Re-assessment

Oral examination = 100%

Learning and teaching methods and delivery

Weekly contact

2.5 lectures (x 10 weeks), 1 tutorial (x 10 weeks)

Scheduled learning hours

24

The number of compulsory student:staff contact hours over the period of the module.

Guided independent study hours

276

The number of hours that students are expected to invest in independent study over the period of the module.

Intended learning outcomes

  • Work with fields and field extensions, construct the field obtained by adjoining a root of a polynomial, and produce theoretical arguments (proofs) about the resulting extension
  • Define what is meant by the degree of a field extension, what it means for a field extension to be algebraic, simple, normal, and separable, and be able to produce theoretical arguments concerning these objects
  • Work with finite fields and theorems concerning their existence, uniqueness (for each order up to isomorphism), and the fact that their multiplicative group is cyclic, and use these properties to solve problems
  • Define the Galois group of a field extension, define the Galois correspondence between the intermediate fields and the subgroups of the Galois group
  • State the Fundamental Theorem of Galois Theory and be able to use it to solve problems about field extensions and Galois groups
  • Define what is meant by a radical extension, to use this concepts in the context of solving polynomial equations, and to construct polynomials that cannot be solved by radicals