Some Homework Hints

Spend time studying for the midterm in addition to doing the homework. To point you in the right direction on the homework problems, here are some hints.

(1)

Use the quadratic reciprocity law, just like in the above examples.

(2)

Use the quadratic reciprocity law.

(3)

Relate the statement for $n=3$ to the statement for $n>3$.

(4)

Write down an element of $(\mathbb{Z}/p^2\mathbb{Z})^*$ that looks like it might have order $p$, and prove that it does. Recall that if $a, b$ have orders $n, m$, with $\gcd(n,m)=1$, then $ab$ has order $nm$.

(5)

(6)

Replace $\sum\left(\frac{a}{p}\right)$ by $\sum \left(\frac{ab}{p}\right)$ and use that $\left(\frac{ab}{p}\right)=\left(\frac{a}{p}\right)\cdot \left(\frac{b}{p}\right)$.

(7)

Write a little program.