1 |
Jan 12 |
Introduction; Fundamental Theorem of Arithmetic |
1.1 |
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- |
- |
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. |