HW 6 due Wed October 12, 2016

  • 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.
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HW5, due Wed Oct 5, 2016

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)

HW4, due Wed Sep 28, 2016

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.

 

HW3, due Wednesday Sep 21, 2016

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.)