MAT210B (Topics in Algebra) Winter 2007
Instructor: Mark Opmeer
Office: MSB 3147
Time and place of lectures: MWF 210 --- 300PM in PHYGEO 140
Textbook: Algebraic Number Theory and Fermat's Last Theorem by Ian Stewart and David Tall.
A good (free!) alternative is provided by the course notes of J.S. Milne.
Something that is very nice to at least glance through is Dedekind's theory of algebraic integers from 1877 (available in a 1996 English translation at Shiels Library).
Prerequisites: undergraduate algebra
Course content: The main topic of the course is Kummer's proof of Fermat's Last Theorem for regular prime exponents. This proof brings together various notions from abstract algebra (such as groups, rings, fields, vector spaces, modules and ideals) in a very nice way to (partially) solve this famous number theory problem.
In student lectures some other topics in algebra will be discussed.
Grading: To get a grade you have to give a 25 minute presentation in class on an algebra topic. In addition you have to convince me in some way that you have mastered the material from the lectures and from the book. The usual way to do this would be to do the take-home exam and an oral exam based on the take-home exam you hand in. Other ways are also possible though.
Schedule:
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In the first lecture(s) I will try to convince you that the study of cyclotomic fields is a natural approach to Fermat's Last Theorem.
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In the second part of the course we will consider some issues raised about cyclotomic fields in part I in the more general context of number fields.
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Using what we learned in part II and assuming Kummer's Lemma on units we give a proof of Fermat's Last Theorem (for regular prime exponents) in part III.
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In part IV we will look at Kummer's Lemma on units (which is not proven in the textbook). A full proof will probably be out of our reach, but I will certainly indicate some of the ingredients of the proof.
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Part V consists of students presentations.
Suggested topics for student presentations:
Wiles' proof of Fermat's Last Theorem
An algebraic proof of the fundamental theorem of algebra
Quadratic reciprocity
Dirichlet's theorem on primes in arithmetic progressions
Chebotarov's generalization of the above theorem of Dirichlet
The Brouwer fixed point theorem (or 'Hairy ball theorem')
Hilbert's Nullstellensatz
Solution of polynomial equations by radicals
Transcendence of pi and e
The prime number theorem
The Riemann hypothesis
The (partial) take-home exam.