MAU34106 Galois theory

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.
• The module is passed if the overall mark for the module is 40% or more. If the overall mark for the module is less than 40% and there is no possibility of compensation, the module will be reassessed as follows: 
  1) A failed exam in combination with passed continuous assessment will be reassessed by an exam in the supplemental session; 
  2) The combination of a failed exam and failed continuous assessment is reassessed by the supplemental exam; 
  3) A failed continuous assessment in combination with a passed exam will be reassessed by one or more summer assignments in advance of the supplemental 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.