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| Module Availability |
Spring |
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| Assessment Pattern |
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Unit(s) of Assessment
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Weighting Towards Module Mark( %)
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Coursework: One take-home assignment and one class test
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25
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Examination: 2 hours, unseen
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75
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| Module Overview |
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.
MMath students will be expected to formulate abstract algebraic reasoning at greater length. |
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| Prerequisites/Co-requisites |
Required: MAT2005 Algebra and Codes
Desirable: MAT2006 Groups and Symmetry
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| Module Aims |
The course will aim to show the power of abstract algebra to produce practical and applicable results. |
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| Learning Outcomes |
On completion students should:
(1) have a deeper appreciation of algebraic structures and of the power of linking different structures, using knowledge of the simpler to gain insights into the more complicated;
(2) be able to construct proofs similar to those met in the course;
(3) be able to evaluate the degree of finite field extensions and apply this to geometric examples;
(4) be able to evaluate specific Galois groups and relate their structure to that of field extensions and the solvability of polynomial equations;
(5) understand the structure of finite fields and find Galois groups of extensions of such fields. |
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| Module Content |
Theory of polynomials. Solution of cubic and quartic equations.
Field extensions, degree of an extension, geometric constructions.
Normal and separable extensions, field automorphisms, the Galois correspondence.
Solvable groups. Conditions for solvability of polynomial equations by radicals.
Finite fields. Galois groups of extensions and polynomials over finite fields.
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| Methods of Teaching/Learning |
Teaching is by lectures and tutorials, 3 hours per week for 11 weeks.
Learning takes place through lectures, tutorials, directed reading, exercises and class tests. |
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| Selected Texts/Journals |
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Recommended:
Ian Stewart : Galois Theory, Chapman & Hall / CRC (2003), ISBN 1584883936
Also useful:
Joseph Rotman : Galois Theory (2nd Edition), Springer-Verlag (1998), ISBN 0387985417.
D.J.H. Garling : A Course in Galois Theory, CUP (1986), ISBN 0521312493.
S. Roman : Field Theory, Springer-Verlag (1995), ISBN 0387944079.
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| Last Updated |
September 10 |
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