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Up: Year 4
Previous: MC446 Algebraic Topology
MC449 Galois Theory
| Credits: 20 |
Convenor: Prof. G. Robinson |
Semester: 2 |
| Prerequisites: |
essential: MC344, MC241, MC242 |
desirable: MC341 |
| Assessment: |
Regular coursework: 20% |
Three hour exam: 80% |
| Lectures: |
36 |
Classes: |
12 |
| Tutorials: |
none |
Private Study: |
102 |
| Labs: |
none |
Seminars: |
none |
| Project: |
none |
Other: |
none |
| Total: |
150 |
|
|
Explanation of Pre-requisites
It will be assumed that the student has some familiarity
with the basic properties of groups, and knows the
beginnings of the theory of field extensions.
It would be desirable, but is not essential, for the student
to have seen the definition of a solvable group.
Course Description
Galois theory is one of the first examples of
methods from one branch of Mathematics being applied
to solve problems in an apparently completely different area,
something which has become relatively common in modern Mathematics.
The original question of the subject was whether or not the
roots of polynomials could be written out explicitly using
only standard arithmetical operations. The theory subsequently
developed gave a beautiful necessary and sufficient condition
for this to be possible, and furthermore described how to
obtain the solutions when it was possible. This course
follows this development of the subject.
Aims
To help the student to develop an appreciation of the rigorous development
of this beautiful subject, and an understanding of the fundamental
results of the subject.
Objectives
To enable the student to :
Understand and define the Galois group of a field
extension, and the Galois group of a polynomial.
Understand and prove the Galois correspondence,
including the relationship between the normal
subject structure of the Galois group and normality
of intermediate extensions.
Understand the definition of a solvable group
and be able to determine whether or not a
group of reasonable size is solvable.
Appreciate the significance of the Galois group
of a polynomial as a group of permutations of the roots.
Prove that the alternating group of degree at least 5
is simple.
Understand the definition of a radical extension and
prove that such an extension has a solvable Galois
group.
Understand Lagrange resolvents and their use
in proving that a finite separable normal extension with a solvable
Galois group is a radical extension.
Understand that the symmetric group is the Galois group
of the general polynomial.
Be able to construct polynomials whose
Galois group is not solvable.
Transferable Skills
This course should assist the student to develop skills
of careful and rigorous Mathematcial writing and
expression. It could also serve as preparation
for s student wishing to embark on research in
Pure Mathematics, as it should convey an appreciation
of a common strand of development in modern
Mathematics.
Syllabus
Automorphisms of field extensions. The Galois group
of a finite extension. Normal extensions. Separability.
The Galois correspondence. The Galois group of a polynomial, viewed
as a group of permutations of the roots. Transitivity
of the Galois group of an irreducible polynomial on
the roots. Composition series for groups and characterization
of solvable groups in terms of these. Alternating groups
and proof of their simplicity in degree greater than 4.
Radical extensions and the solvability of their Galois
groups. Lagrange resolvents and the proof that a radical
normal extension has a solvable Galois group.
The symmetric group as the Galois group of the ``general''
polynomial. Explicit determination of the general solution
of the general cubic and quartic. Examples of polynomials
which are not solvable by radicals.
Reading list
Essential:
Recommended:
I. Stewart,
Galois Theory,
Chapman and Hall.
Details of Assessment
The coursework will consist of regularly assigned
exercise sheets. The June examination will have 6 questions,
it being (theoretically) possible to obtain full marks by answering 4 of
them.
Next: MC460 Statistical Inference
Up: Year 4
Previous: MC446 Algebraic Topology
Roy L. Crole
10/22/1998