Let $\mathbb{F}_q$ be a finite field of characteristic $p \neq 2,5$ .
What I've shown so far :
$x \in\mathbb{F}_q^* $ is a root of $\Phi_5 = X^4 + X^3 +X^2+X+1$ if and only if $x$ is of order 5 in $\mathbb{F}_q^* $.
$\Phi_5$ has a root in $\mathbb{F}_q$ if and only if $q-1$ is divisible by 5. If that is the case, and if $x \in \mathbb{F}_q$ is a root of $\Phi_5$, all the roots of $\Phi_5$ are $x, x^2, x^3$ and $x^4$ and therefore $\Phi_5$ is splitted over $\mathbb{F}_q$.
If $x$ is a root of $\Phi_5$ in a field extension of $\mathbb{F}_p$, and if we note $y=x+x^{-1}$, then : $(2y+1)^2 = 4y^2+4y+1 = 4(x^{-2}+2+x^2+x^{-1}+x) +1 = 4x^{-2}\Phi_5(x)+4+1 = 5$
What I'm struggling on :
- Show that 5 is a square in $\mathbb{F}_p$ if and only if $y\in \mathbb{F}_p$ ($y$ defined as above). What can we say about the degree of $x$ in $\mathbb{F}_p$ ?
- Show that in this case $\Phi_5$ is splitted over $\mathbb{F}_{p^2}$ and that $p \equiv \pm 1 \pmod 5$.
- Suppose that $p \equiv - 1 \pmod 5$. What is $x^{p+1}$ worth ? Deduce that $y \in \mathbb{F}_p$