Math 401  Fall 2015 
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.
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; zerodivisors and integral domains; fields; examples  1.4  
Thursday 17 September  Zerodivisors 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 15puzzle  6.4 