Galois Theory

Module code: G1157
Level 6
15 credits in autumn teaching
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