Let $P (X) = X^5 − 6X + 3$.
Prove that it is irreducible over $\mathbb{Q}$.
In the solution I have for this exercice, I have litterally:
P is irreducible over $\mathbb{F}_5$, therefore, by Gauss Lemma, it is irreducible over $\mathbb{Q}$.
I am lacking the necessary knowledge about irreducibility by reducing modulo p. All I know is that there is a bijection between roots of P in $\overline{\mathbb{Q}}$ and the roots of $\overline{P}$ in $\overline{\mathbb{F}_p}$.
I also know this generalization of Gauss Lemma: in a factorial ring A, with its fraction field K, a primitive $P$ is irreducible in $A[X] \Leftrightarrow $ $P$ is irreducible in K[X].
How does this apply to P above?
Thank you for any directions or help.