GALOIS THEORY - 2021/2
Module code: MAT3011
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Galois Theory applies the principles of algebraic structure to questions about the solvability of polynomial equations. The feasibility of certain geometrical constructions is also considered.
FISHER David (Maths)
Number of Credits: 15
ECTS Credits: 7.5
Framework: FHEQ Level 6
JACs code: G100
Module cap (Maximum number of students): N/A
Prerequisites / Co-requisites
MAT2048 Groups and Rings
Indicative content includes:
- Theory of polynomials. Criteria for irreducibility.
- Solution of cubic and quartic equations.
- Field extensions. The degree of an extension. The Tower law.
- Geometric constructions.
- Field automorphisms and their properties.
- The Galois correspondence and the fundamental theorem.
- Solvable groups. Conditions for solvability of polynomials by radicals.
|Assessment type||Unit of assessment||Weighting|
|School-timetabled exam/test||IN-SEMESTER TEST (50 MINS)||20|
|Examination||EXAMINATION (2 HOURS)||80|
The assessment strategy is designed to provide students with the opportunity to demonstrate their ability to
· construct and interpret mathematical arguments in the context of this module;
· display subject knowledge by recalling key definitions and results;
· apply the techniques learnt to both routine and unfamiliar problems.
Thus, the summative assessment for this module consists of:
· One two-hour examination at the end of Semester 1, worth 80% of the module mark.
· In-semester test, worth 20% of the module mark.
Formative assessment and feedback
Students receive written comments on their marked coursework assignments. Maple TA may be used in assignments to provide instant grading and immediate feedback. Verbal feedback is provided in lectures and office hours.
- review the theory of groups, rings, fields and polynomials.
- develop and apply the theory of field extensions and Galois groups.
- show the power of abstract algebra to produce practical and applicable results.
|1||Demonstrate a deeper appreciation of algebraic structures and of the power of linking different structures||KCT|
|2||Solve cubic and quartic equations, understanding the limitations of the methods||KC|
|3||Evaluate the degree of finite field extensions and apply this to algebraic and geometric examples||KC|
|4||Evaluate specific Galois groups and relate their structure to that of field extensions and to the solvability of polynomial equations .||KC|
|5||Construct simple proofs similar to those encountered in the module.||KCT|
C - Cognitive/analytical
K - Subject knowledge
T - Transferable skills
P - Professional/Practical skills
Overall student workload
Independent Study Hours: 117
Lecture Hours: 33
Methods of Teaching / Learning
The learning and teaching strategy is designed to provide:
- An awareness of the applicability of abstract algebra to classical problems.
- Knowledge of the historical context in which the subject material was developed.
- Experience of the methods used to interpret, understand and solve problems in Galois theory.
The learning and teaching methods include:
- Three 50-minute lectures per week for eleven weeks, some being used as tutorials, problem classes and in-semester tests.
- Online notes supplemented by additional examples in lectures.
- Two unassessed coursework assignments, marked and returned.
- Personal assistance given to individuals and small groups in office hours.
Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.
Upon accessing the reading list, please search for the module using the module code: MAT3011
Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2021/2 academic year.