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Homework 4: Primitive Roots and
Quadratic Reciprocity
DUE WEDNESDAY, OCTOBER 17

William Stein

Date: Math 124 $\quad$ HARVARD UNIVERSITY $\quad$ Fall 2001

(1 point) Why do you think that quadratic reciprocity is so cool?

1.: (2 points) Calculate the following symbols by hand: $\left(\frac{3}{97}\right)$ , $\left(\frac{5}{389}\right)$ , $\left(\frac{2003}{11}\right)$ , and $\left(\frac{5!}{7}\right)$ .
2.: (3 points) Prove that $\displaystyle \left(\frac{3}{p}\right) = \begin{cases}\hfill 1 & \text{ if }p\equiv 1, 11\pmod{12},\\ -1 & \text{ if }p\equiv 5, 7\pmod{12}. \end{cases}$
3.: (3 points) Prove that there is no primitive root modulo for any $n\geq 3$ .
4.: (6 points) Prove that if is a prime, then there is a primitive root modulo .
5.: (5 points) Use the fact that $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic to give a direct proof that $\left(\frac{-3}{p}\right)=1$ when $p\equiv 1\pmod{3}$ . [Hint: There is an $c\in (\mathbb{Z}/p\mathbb{Z})^*$ of order . Show that .]
6.: (6 points) If $p\equiv 1\pmod{5}$ , show directly that $\left(\frac{5}{p}\right)=1$ by the method of Exercise 5. [Hint: Let $c\in (\mathbb{Z}/p\mathbb{Z})^*$ be an element of order . Show that , etc.]
7.: (4 points) For which primes is $\displaystyle \sum_{a=1}^{p-1} \left(\frac{a}{p}\right)=0$ ?
8.: (4 points) Artin conjectured that the number of primes $p\leq x$ such that is a primitive root modulo is asymptotic to $C\pi(x)$ where $\pi(x)$ is the number of primes $\leq x$ and is a fixed constant called Artin's constant. Using a computer, make an educated guess as to what should be, to a few decimal places of accuracy. Explain your reasoning. (Note: Don't try to prove that your guess is correct.)