How does the polynomial $X^{p-1}+1$ split over $\mathbb{F}_p$

Question

Is there a well-known formula for the irreducible factors of the polynomial $X^{p-1}+1$ over $\mathbb{F}_p$ where $p$ is an odd prime? Thanks in advance.

Answer 1

Its irreducible factors are $$X^2-r$$ (at least when $p$ is odd) as $r$ runs through the quadratic non-residues modulo $p$.

This follows, say, from Euler's criterion for the Legendre symbol.

Answer 2

You can write for $p$ odd prime :

$$ X^{p-1} + 1 = \prod_{ a , (\frac{a}{p})=-1}(X^2-a) \mod p$$

With :

$$ (\dfrac{a}{p})=-1 $$ denotes the Legender symbol that is $-1$ if $a$ is not a quadratic residues $ \mod p$

Then I used Euler criterion :

$$ (\dfrac{a}{p})=a^{\frac{p-1}{2}} \mod p $$