MAS442 Galois Theory

Semester 2, 2017/18 10 Credits
Lecturer: Dr Kirill Mackenzie Home page Reading List
Aims Outcomes Teaching Methods Assessment Full Syllabus

Given a field K (as studied in MAS333/438) one can consider the group G of isomorphisms from K to itself. In the cases of interest, this is a finite group, and there is a tight link (called the Galois correspondence) between the structure of G and the subfields of K. If K is generated over the rationals by the roots of a polynomial f(x), then G can be identified as a group of permutations of the set of roots. One can then use the Galois correspondence to help find formulae for the roots, generalising the standard formula for the roots of a quadratic. It turns out that this works whenever the degree of f(x) is less than five. However, the fifth symmetric group lacks certain group-theoretic properties that lie behind these formulae, so there is no analogous method for solving arbitrary quintic equations. The aim of this course is to explain this theory, which is strikingly rich and elegant.

Prerequisites: MAS333 (Fields) or MAS438 (Fields)
No other modules have this module as a prerequisite.


Outline syllabus

  • Review of fields and other background
  • Homomorphisms and field extensions
  • Splitting fields
  • Extending homomorphisms; normal field extensions; Galois groups
  • Examples involving extensions of small degree
  • Cyclotomic fields and their Galois groups
  • Finite fields (if time permits)
  • The Galois correspondence
  • Cubics and quartics
  • Extension by radicals and solvability.



Aims

  • To explain the general theory of homomorphisms between fields.
  • To explain the definition of Galois groups, and to compute them for cyclotomic extensions, and various extensions of small degree.
  • To explain the Galois correspondence, and use it to reduce various questions in field theory to easier questions about finite groups.
  • To study splitting fields and Galois theory for cubics and quartics, and to explain how they lead to algorithms for finding roots.

Learning outcomes

  • understand the main ideas of Galois theory.
  • compute Galois groups for fairly simple field extensions, including cyclotomic extensions.
  • compute Galois groups for fairly simple polynomials.
  • use the Galois correspondence to solve problems about the structure of fields.

Teaching methods

Lectures, problem solving


20 lectures, no tutorials

Assessment

One formal 2.5 hour written examination. All questions compulsory.

Full syllabus

1. Fields and vector spaces

Rapid review of MAS333/438 and other background.
2. Ideals and quotient rings
Mostly revision.
3. Polynomials over fields
Mostly revision.
4. Adjoining roots
Adjoining roots, and properties of the resulting field extensions. Splitting fields.
5. Extending homomorphisms
Dedekind's lemma. Inequalities for numbers of extensions. Normal extensions, and criteria for normality. Galois groups.
6. Some extensions of small degree
Selected examples such as Q(√2,√3) and Q(√{3+√5}).
7. Cyclotomic extensions
Definition and properties of cyclotomic polynomials and their splitting fields. Automorphisms of cyclotomic fields. Examples of subfields of cyclotomic fields.
8. Finite fields
There is a field of each prime-power order, and it is unique up to isomorphism. Galois groups of extensions of finite fields. (May be omitted if time is short.)
9. The Galois correspondence
For any normal extension L/K, there is a bijection between subgroups of the Galois group G(L/K) and intermediate fields between K and L. Finer properties of this bijection. Examples of using the correspondence to derive facts about field structure.
10. Cubics
Galois groups of splitting fields of cubics. The discriminant criterion for whether the group is A3 or Σ3.
11. Quartics
Subgroups of the symmetric group Σ4. Corresponding facts about quartics, including the resolvent cubic.
12. Cyclic extensions
Splitting fields for polynomials of the form xn−a, and their Galois groups.
13. Solvability and extension by radicals
Basic theory of solvable groups. Polynomials are solvable by radicals if and only if the corresponding Galois groups are solvable. Quintics are usually not solvable by radicals.

Reading list

Type Author(s) Title Library Blackwells Amazon
B Edwards Galois theory 512.81 (E) Blackwells Amazon
B Escofier Galois theory 512.81 (E) Blackwells Amazon
B Garling A course in Galois theory 512.81 (G) Blackwells Amazon
B Rotman Galois theory 512.81 (R) Blackwells Amazon
B Stewart Galois theory 512.43 (S) Blackwells Amazon
C King Beyond the quartic equation 512.3 (K) Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.