MAU34101 Galois theory
| Module Code | MAU34101 |
|---|---|
| Module Title | Galois theory |
| Semester taught | Semester 1 |
| ECTS Credits | 5 |
| Module Lecturer | Prof. Nicolas Mascot |
| Module Prerequisites | MAU22102 Fields, rings and modules |
Assessment Details
- This module is examined in a 2-hour examination at the end of Semester 1.
- Continuous assessment contributes 20% towards the overall mark.
- Any failed components are reassessed, if necessary, by an exam in the reassessment session.
Contact Hours
11 weeks of teaching with 3 lectures per week.
Learning Outcomes
On successful completion of this module, students will be able to
- State and explain relationships between properties of field extensions and properties of their automorphism groups.
- Explicitly construct finite fields of small orders.
- Determine Galois groups of polynomials of small degree.
- Illustrate applications of Galois theory on specific examples.
Module Content
- Polynomial rings: UFD and PID property, Gauss lemma, Eisenstein's criterion.
- Algebraic field extensions: Tower law, ruler and compass constructions.
- Splitting fields and their properties, classification of finite fields.
- Normal and separable extensions, primitive element theorem.
- Galois extension, Galois correspondence, fundamental theorem of algebra.
- Algorithm for computing the Galois group of a given polynomial.
- Solvability by radicals. Cyclic, abelian and solvable field extensions.
- Abel's theorem for equations of degree five.
- Abelian and cyclotomic extensions, Kronecker-Weber theorem.