Math 401 Fall 2015

Syllabus

This is a tentative schedule of lecture topics, reading and homework assignments, and exams. You are expected to read and study the text as we go along. Whether you study the text before or after the lecture is up to you; the best approach is to do both. Assignments are due at the beginning of class on the indicated due date.

The homework assignments and handouts are PDF files which may be viewed or printed with a recent version of the free Adobe Acrobat Reader.

I stress that the exam times and assignment due dates indicated here are subject to change.

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Date Topics Text Assignments
Tuesday 25 August Introduction, conventions for notation, sets, truth tables 1.1, A.1 Assignment 1
(due 1 September)
Thursday 27 August Induction, binomial theorem, infinity of primes, division algorithm 1.1, 1.2
Tuesday 1 September g.c.d., Euclidean algorithm, fundamental theorem of arithmetic 1.2, 1.3 Assignment 2
(due 8 September)
Thursday 3 September Congruence, equivalence relations in general, 1.3, A.3
Tuesday 8 September Congruence, Fermat Little Theorem, Euler generalization, Chinese Remainder Theorem 1.3 Assignment 3
(due 17 September)
Thursday 10 September Arithmetic in Z mod m and rings; units; cryptography and the RSA cryptosystem 1.3, 1.4; handout
Tuesday 15 September Solving congruences, systems of congruences and quadratic congruences; zero-divisors and integral domains; fields; examples 1.4
Thursday 17 September Zero-divisors and integral domains; fields; examples 1.4, 3.1 Assignment 4
(due 24 September)
Tuesday 22 September Polynomial rings; division algorithm for monic polynomials; irreducibility of polynomials, g.c.d. 3.2
Thursday 24 September Euclidean algorithm for polynomials over a field, unique factorization of polynomials over a field Assignment 5
(due 1 October)
Tuesday 29 September Complex numbers; other fields 2.3, 3.2
Thursday 1 October Roots and field extensions 3.2, 3.3 Assignment 6
(due 15 October)
Tuesday 6 October Algebraic elements; minimal polynomial; irreducibility of rational polynomials by undetermined coefficients 3.3
Thursday 8 October Gauss's lemma; Reduction mod p, Eisenstein criterion Assignment 7
(due 22 October)
Fall Break
Thursday 15 October Fields as congruence classes of polynomials 4.1
Tuesday 20 October Homomorphisms 4.1
Thursday 22 October Hour Exam Assignment 8
Tuesday 27 October Ring isomorphisms, ideals 4.2 Midterm solutions
Thursday 29 October Fundamental homomorphism theorem 4.2 Assignment 9
(due November 5)
Tuesday 3 November Numbers constructible with straightedge and compass 5.1
Thursday 5 November Vector spaces and dimension 5.1 Assignment 10
(due 12 November)
Tuesday 10 November Dimension of extensions and applications; impossibility of angle trisection with straightedge and compass 5.2
Thursday 12 November Groups: definition and examples 6.1 Assignment 11
(due 22 November)
Tuesday 17 November Subgroups, homomorphisms, isomorphisms 6.2
Thursday 19 November Normal subgroups, quotient groups 6.2
Tuesday 24 November The structure of the symmetric group 6.3 Assignment 12
(due 3 December)
Thanksgiving Break
Tuesday 1 December The symmetric group and the attack on Enigma Handout
Thursday 3 December The symmetric group and the 15-puzzle 6.4

Final Exam:
Wednesday 9 December, 7-10PM (Section 1)
Tuesday 8 December, 7-10PM (Section 2)



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