Factor $x^5-1$ into irreducibles in $\mathbb{F}_p[x]$ I have to factor the polynomial $f(x)=x^5-1$ in $\mathbb{F}_p[x]$, where $p \neq 5$ is a generic prime number.
I showhed that, if $5 \mid p-1$, then $f(x)$ splits into linear irreducible.
Now I believe that, if $5 \nmid p-1$ but $5 \mid p+1$, then $f(x)$ splits into three irreducible polynomial, one of degree $1$ and two of degree $2$.
Oterwise, if $5 \nmid p-1$ and $5 \nmid p+1$ (so $p \mid p^2+1$), then $f(x)$ splits into two irreducible factors, one of degree $1$ and one of degree $4$.
How can I prove this statements (if I'm right, well...)? Thank you!

Hints (for you to understand, complete/prove)
Over any field
$$x^5-1=(x-1)(x^4+x^3+x^2+ x +1)$$
Now, the roots of the second factor above are the roots of unity of order $\,5\,$ different from 1 itself. Since these roots (including $\;1\;$) form a cyclic group of order$\; 5\;$, we want to now when $\,5\mid p-1\,$ . In these cases we see the above pol. will split in different linear factors, for example:
$$p=11\;\implies x^5-1=(x-1)(x-4)(x-5)(x-9)(x-3)\in\Bbb F_{11}[x]$$
Also note that $\,a\neq 1\,$, otherwise the above equations have no solution (why?), so the other option for $\,b\,$ still's open. Try now to take it from here.
Note that $\,4=4^1\;,\;5=4^2\;,\;9=4^3\;,\;3=4^4\,$
It though can be the above quartic splits as the product of two irreducible quadratics:
$$x^4+x^3+x^2+x+1=(x^2+ax+b)(x^2+cx+d)=$$
$$=x^4+(a+c)x^3+(d+ac+b)x^2+(ad+bc)x+bd$$
Comparing corresponding coefficients we get the equations
$$\begin{align*}a+c&=1\\ac+b+d&=1\\ad+bc&=1\\bd&=1\end{align*}$$
Thus
$$(1)+(3)\;\;a=1-c\;,\;d=\frac1b\implies (3)\;\;\frac ab+b(1-a)=1\implies$$
$$(1-a)b^2-b+a=0\;:\;\;\Delta=1-4a(1-a)=4a^2-4a+1=(2a-1)^2\implies$$
$$b_{1,2}=\frac{1\pm (2a-1)}{2(1-a)}=\begin{cases}\frac a{1-a}\\{}\\\;\;\;\;1\end{cases}$$
Now
$$b=1\implies d=1\;,\;\;ac=-1\;,\;c=1-a\implies a(1-a)=-1\implies$$
$$ a^2-a-1=0\implies a=\frac{1\pm\sqrt 5}2$$
Note that this choice requires $\,\sqrt 5\in\Bbb F_p\iff \binom5p=1\,$ ,but by quadratic reciprocity
$$1=\binom5p=\binom p5\iff p=1,4\pmod 5$$
