Math 480 (Spring 2007): Homework 6

Due: Monday, May 7


There are 5 problems. Each problem is worth 6 points and parts of multipart problems are worth equal amounts. You may work with other people and use a computer, unless otherwise stated. Acknowledge those who help you.

  1. Calculate the following Legendre symbols by hand (you may use the quadratic reciprocity law): $ \left(\frac{4}{10007}\right), \left(\frac{3}{37}\right), \left(\frac{5}{11}\right), \left(\frac{5!}{7}\right)$ .

  2. Let $ G$ be an abelian group and let $ n$ be a positive integer.
    1. Prove that the map $ \varphi :G\to G$ given by $ \varphi (x) = x^n$ is a group homomorphism.
    2. Prove that the subset $ H$ of $ G$ of squares of elements of $ G$ is a subgroup.

  3. Give an example of an abelian group $ G$ and two distinct subgroups $ H$ and $ K$ both of index $ 2$ . Note that $ G$ will not be cyclic.

  4. (*) Prove that for any $ n\in\mathbb{Z}$ the integer $ n^2+n+1$ does not have any divisors of the form $ 6k-1$ . (Hint: First reduce to the case that $ 6k-1$ is prime, by using that if $ p$ and $ q$ are primes not of the form $ 6k-1$ , then neither is their product. If $ p=6k-1$ divides $ n^2+n+1$ , it divides $ 4n^2+4n+4 = (2n+1)^2+3$ , so $ -3$ is a quadratic residue modulo $ p$ . Now use quadratic reciprocity to show that $ -3$ is not a quadratic residue modulo $ p$ .)

  5. For each of the following equations, either find all integer solutions with $ 0\leq x <p$ or prove that no solutions exist:
    1. $ x^2 + 2x + 3 \equiv 0 \pmod{7}$ , where $ p=7$ .
    2. $ x^2 - x + 7 \equiv 0 \pmod{11}$ , where $ p=11$ .
    3. $ x^2 + x + 1 \equiv 0 \pmod{2}$ , where $ p=2$ .
    4. $ x^2 - 3 \equiv 0 \pmod{389}$ , where $ p=389$ .
    5. $ x^2 + x + 1 \equiv 0 \pmod{3}$ , where $ p=3$ .
    6. $ 2x^2 + 3x - 2 \equiv 0 \pmod{5}$ , where $ p=5$ .



William 2007-05-02