# Kevin Milans : Teaching : Spring 2016 Math645

Kevin Milans (milans@math.wvu.edu)
Office: Armstrong Hall 408H
Office Hours: MTuTh 11:30am-12:30pm, and by appointment
Class Meetings: TuTh 2:30pm-3:45pm in Hodges Hall 302

Home | Course Syllabus (PDF) | Homework

### Course Schedule

 No. Date Class Summary Section(s) Quiz Quiz Soln Comments 1 Jan 12 Introduction; Fundamental Theorem of Arithmetic 1.1 - - - 2 Jan 14 Polynomial rings 1.2,1.3 - - HW1 assigned. 3 Jan 19 Principal Ideal Domains 1.3 - - - 4 Jan 21 F.T.A. for PID's; Gaussian integers 1.3,1.4 - - - 5 Jan 26 Infinitely many primes; square-free; Mobius inversion 2.1,2.2 - - - 6 Jan 28 Sum of 1/p over primes p; density of primes 2.3,2.4 - - HW1 due. HW2 assigned. 7 Feb 2 Congruences; quotient rings 3.2,3.3 - - - 8 Feb 4 Chinese Remainder Theorem 3.4 - - - 9 Feb 9 Structure of the group of units modulo m 4.1 - - - 10 Feb 11 Structure of the group of units modulo m II 4.1 - - HW2 due. HW3 assigned. 11 Feb 16 Quadratic residues, nonresidues 5.1 - - - 12 Feb 18 Quadratic reciprocity: applications 5.2 - - - 13 Feb 23 Quadratic reciprocity: alternative forms 5.2 - - - 14 Feb 25 Algebraic numbers and integers 6.1 - - HW3 due. HW4 assigned. 15 Mar 1 Algebraic numbers and integers II 6.1 - - - 16 Mar 3 Quadratic character of 2 6.2 - - - 17 Mar 8 Proof of QRL; sign of Quadratic Gauss Sum 6.3,6.4 - - - 18 Mar 10 Finite fields 7.1 - - HW4 due. HW5 assigned. 19 Mar 15 Finite fields; Field extensions 7.1,7.2 - - - 20 Mar 17 Number of irreducible polynomials 7.2 - - HW4 due. HW5 assigned. 21 Mar 29 Quadratic reciprocity via finite fields 7.3 - - - 22 Mar 31 Characters; Gauss Sums 8.1,8.2 - - HW5 extended to next Thursday. 23 Apr 5 Gauss Sums; Jacobi Sums 8.2,8.3 - - - 24 Apr 7 Jacobi Sums 8.3 - - HW5 due. 25 Apr 12 Counting solutions to the eqn x^n + y^n = 1; multi-arg Jacobi sums 8.4,8.5 - - - 26 Apr 14 Multi-argument Jacobi sums 8.5 - - HW6 assigned; a few problems to be added. 27 Apr 19 Multi-argument Jacobi sums and Gauss sums; counting solutions to x12 + ... + xn2 = 1 8.5, 8.6 - - - 28 Apr 21 Cubic Reciprocity I: primes in Z[ω] 9.1 - - - 29 Apr 26 Cubic Reciprocity II: residue rings, cubic character 9.2,9.3 - - - 30 Apr 28 Cubic Reciprocity III: proof in the case of one rational and one complex prime 9.4 - - HW6 due.

milans@math.wvu.edu