# Mathematics

## Galois Theory

Module code: G1157
Level 6
15 credits in autumn semester
Teaching method: Lecture
Assessment modes: Unseen examination, Coursework

A quadratic equation in one variable has a formula for its solutions. So do cubic and quartic equations, whereas a general quintic has no such formula. The theory of field equations and its connection to the theory of groups explains this.

The syllabus will include:

• consideration of the historic problems
• quadratic equations, complex roots of 1, cubic equations, quartic equations
• insolvability of the quintic
• ruler and compass constructions, squaring the circle, duplicating the cube
• field extensions
• applications to ruler-and-compass constructions
• normal extensions
• application to finite fields, splitting fields
• Galois group of polynomials
• application to x5 - 1 = 0
• fundamental theorem of Galois Theory
• Galois group for cubic polynomial
• solutions of equations in radicals and soluble groups.

### Module learning outcomes

• Investigate and solve a cubic equation
• investigate and solve a quartic equation
• investigate and solve a quartic equation
• accurately perform ruler and compass constructions