Undergraduate Programme and Module Handbook 2011-2012 (archived)
Module MATH3041: GALOIS THEORY III
Department: Mathematical Sciences
MATH3041:
GALOIS THEORY III
Type |
Open |
Level |
3 |
Credits |
20 |
Availability |
Available in 2011/12 |
Module Cap |
None. |
Location |
Durham
|
Prerequisites
Corequisites
Excluded Combination of Modules
Aims
- To introduce the way in which the Galois group acts on the field
extension generated by the roots of a polynomial, and to apply this to
some classical ruler-and-compass problems as well as elucidating the
structure of the field extension.
Content
- Field Extensions: Algebraic and transcendental extensions,
splitting field for a polynomial, normality,
separability.
- Results from Group Theory: Normal subgroups, quotients,
soluble groups, isomorphism theorems.
- Groups acting on fields: Dedekind's lemma, fixed field,
Galois group of a finite extension, definition of Galois extension,
fundamental theorem of Galois theory.
- Galois Group of Polynomials: Criterion for solubility in
radicals, cubics, quartics, 'general polynomial', cyclotomic
polynomials.
- Ruler and Compass Constructions: definition, criterion for
constructability, impossibility of trisecting angle, etc.
- Further Topics.
Learning Outcomes
- By the end of the module students will: be able to solve
novel and/or complex problems in Galois Theory.
- have a systematic and coherent understanding of theoretical
mathematics in the field of Galois Theory.
- have acquired a coherent body of knowledge of these subjects
demonstrated through one or more of the following topic areas:
Algebraic field extensions, properties of normality and
separability.
- Properties of Galois correspondence.
- Criterion of solvability of polynomial equation in
radicals.
- Non-solvability of general polynomial equation in degrees
> 5.
- Classification of finite fields.
- Construction of irreducible polynomials with coefficients in
finite fields.
- In addition students will have specialised mathematical
skills in the following areas which can be used with minimal guidance:
Abstract reasoning.
Modes of Teaching, Learning and Assessment and how these contribute to
the learning outcomes of the module
- Lectures demonstrate what is required to be learned and the
application of the theory to practical examples.
- Assignments for self-study develop problem-solving skills and
enable students to test and develop their knowledge and
understanding.
- Formatively assessed assignments provide practice in the
application of logic and high level of rigour as well as feedback for
the students and the lecturer on students' progress.
- The end-of-year examination assesses the knowledge acquired
and the ability to solve predictable and unpredictable
problems.
Teaching Methods and Learning Hours
Activity |
Number |
Frequency |
Duration |
Total/Hours |
|
Lectures |
40 |
2 per week for 19 weeks and 2 in term 3 |
1 Hour |
40 |
|
Preparation and Reading |
|
|
|
160 |
|
Total |
|
|
|
200 |
|
Summative Assessment
Component: Examination |
Component Weighting: 100% |
Element |
Length / duration |
Element Weighting |
Resit Opportunity |
three hour written examination |
|
100% |
|
Four written assignments to be assessed and
returned. Other assignments are set for self-study and complete solutions
are made available to students.
■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University