- MNT 7.2
- MNT 7.3
- MNT 7.16
- MNT 7.18
- Let F3(i) be the field with 9 elements, where i is an element whose square is -1. Find all the generators of the group of units of F3(i).
- Find all the generators of the group of units of Z/11Z.
Here is a link to the Blum-Blum-Shub paper.
Study the proof of quadratic reciprocity in MNT, as well as Rousseau’s proof based on the Chinese Remainder theorem.
- MNT Problem 5.7
- MNT Problem 5.9
- MNT Problem 5.12
- MNT Problem 5.17
- MNT Problem 5.22
- MNT Problem 5.38 (first part, but not the generalization)
There will be an midterm exam on Wednesday, October 19, 2016 during our regularly scheduled class.
I recommend two particular proofs of quadratic reciprocity.
Exercises Chapter 3, MNT page 37
- problem 3.5,
- problem 3.10,
- problem 3.14
- problem 3.16
Exercises Chapter 4, MNT page 49
- optional/bonus problem 4.22
Exercises Chapter 5, MNT page 63-65
- problem 5.3
- problem 5.16.
Find all the quadratic residues modulo 41. You can use WolframAlpha or other computer programs to find the answer. In WolframAlpha Mod[x,41] computes x modulo 41.
Chapter 2, MNT pages 26-27
- Problem 11
- Problem 13 (Hint: insert a “k” where appropriate in the argument on page 19).
- Problem 15a (Hint: do you recognize this as a Dirichlet product? How is the formula related to the Moebius inversion formula?)
- Problem 26a,b. There is no need to prove convergence of either side. For full credit, it is enough to work the case s=2. (Hint: use the product formula over primes in problem 25 for zeta(s).)
Chapter 3, MNT pages 36-37
- Problem 2
- Problem 4 (Hint: if an equation has no solutions modulo m for some m, then it has no solutions in the integers.)