Let $m,n,c\in\mathbb{N}$ $$S(m,n,c)=\sum_{(x,c)=1, x\in \{1,...,c\}} e^{\frac{2\pi(mx+n\overline{x})}{c}}$$ where $x\overline{x}\equiv 1 $(mod) $c$.
I am working on a problem where I wish to show Kloosterman Sum $S(1,a,p)\neq 0$ where $p$ is a prime number. I came across an answer. However I am not able to understand it properly. Any help is much appreciated. Here is a link to the answer.
https://mathoverflow.net/a/188454/123716
I specifically want to show (2.1) of the answer.
If $S(1,a,p)=\sum_{y=1}^{p-1}e^{\frac{2i\pi}{p}(y+y^*)}$ then it is non-zero in $\Bbb{Z}[e^{2i\pi /p}]\cong \Bbb{Z}[x]/(\Phi_p(x))$ because there is a ring homomorphism $$\Bbb{Z}[e^{2i\pi /p}]\to \Bbb{Z}[x]/(\Phi_p(x))\to \Bbb{Z}[x]/(\Phi_p(x),p)$$ $$=\Bbb{Z}[x]/((x-1)^{p-1},p)\to\Bbb{Z}[x]/(x-1,p)\to \Bbb{F}_p$$ sending $e^{2i\pi/p}$ to $1$ and $S(1,a,p)$ to $-1$.