Just as there is a formula for solving a quadratic equation, there are similar formulae for solving the general cubic and quartic. Galois theory provides a solution to the corresponding problem for quintics --- there is no such formula in this case! Galois theory also enables us to prove (despite regular claims to the contrary) that there is no ruler and compass construction for trisecting an angle. More broadly, the purpose of Galois theory is to study polynomials at a deep level by using symmetries between the roots. This is a pervasive theme in modern mathematics, and Galois theory is traditionally where one first encounters it.

Topics to be covered include:

Galois Theory - fields, field extensions, normal extensions, separable extensions. Revision of group theory, abelian and soluble groups.The main theorem of Galois theory.Solubility of equations by radicals. Finite fields. Cyclotomic fields.

Note: This is an HPC. It emphasises mathematical rigour and proof and continues the development of modern analysis from an abstract viewpoint.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of field extensions and Galois theory and their role in modern mathematics and applied contexts
2. Demonstrate accurate and efficient use of field extensions and Galois theory
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from field extensions and Galois theory
4. Apply problem-solving using field extensions and Galois theory applied to diverse situations in physics, engineering and other mathematical contexts.

Assessment will be based on:

• Assignment 1 (15%; LO 1-4)
• Assignment 2 (15%; LO 1-4)
• Assignment 3 (15%; LO 1-4)
• Assignment 4 (15%; LO 1-4)
• Final exam (40%; LO 1-4)

36 lectures, tutorials by arrangement

A mark of 60 or more in MATH2322.