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 $p=5$ .