How can I prove that $\left( \frac{5}{p} \right) = 1$ given that $p \equiv 1 \pmod{5}$ (where $\left( \frac{p}{q} \right)$ is the Legendre symbol)?

Question

I know how to prove this using the law of quadratic reciprocity, but the book suggests proving $(c^4 + c)^2 + (c^4 + c) - 1 = 0$, using $c \in (\mathbb{Z}/p\mathbb{Z})^*$ such that $c^5 \equiv 1 \pmod{p}$. I'm not how to either show that or how it would help if I could show that.

Any hints would be appreciated, thanks.

Answer 1

Hint: The roots of the polynomial $x^2+x-1$ are $x=\dfrac{-1\pm\sqrt{5}}{2}$.

What do you expect $2(c^4+c)+1$ will equal in $\mathbb{Z}/p\mathbb{Z}$?