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| Module Availability |
Spring Semester |
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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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Exam: 2 hours, unseen
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75%
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| Module Overview |
Galois Theory applies the principles of algebraic structure, as studied in Linear Algebra and Group Theory, to questions about the solvability of polynomial equations. The feasibility of certain geometrical constructions is also considered. |
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| Prerequisites/Co-requisites |
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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 simple 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.
Additionally, MMath students should understand some more advanced topics and be able to formulate abstract algebraic reasoning at greater length. |
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| Module Content |
Groups, rings and fields. Field extensions, degree of extension, geometric constructions.
Finite fields.
Normality and separability, field degrees and group orders, automorphisms.
The Galois correspondence, solvable groups, solutions of equations by radicals.
Additional topics for the MMath course may include: Field homomorphisms and the evaluation map, Fermat's Theorem, inseparable polynomials, the Affine Group |
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| Methods of Teaching/Learning |
Teaching is by lectures and tutorials. Learning takes place through lectures, tutorials, directed reading, exercises and class tests. 3 hours per week for 10 weeks plus directed reading. |
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| Selected Texts/Journals |
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 |
04.11.2008 |
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