Reduction modulo $2$ proves $x^3+17x+391$ irreducible over $\mathbb Q$

Given: The primitive integer polynomial $f=x^3+17x+391$.

If a primitive integer polynomial has leading coefficient nonzero modulo a prime and its reduction is irreducible, then it is irreducible over $\mathbb Q$ (Irreducibility after reduction modulo a prime implies irreducibility over $\mathbb Q$ when the leading coefficient survives).

A cubic over a field is irreducible exactly when it has no root in the field (A polynomial of degree two or three over a field is irreducible exactly when it has no root in the field).

The ring $\mathbb Z/2$ is the quotient $\mathbb Z/2\mathbb Z$ (For every $n\in\mathbb N$, the congruence-class ring $\mathbb Z/n$ is the quotient ring $\mathbb Z/n\mathbb Z$).

An integer is prime when it exceeds $1$ and has no positive divisors other than $1$ and itself (Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$).

Integer divisors of a nonzero integer are bounded in absolute value (If $d \mid a$ and $a \ne 0$ then $d \ne 0$ and $|d| \le |a|$; hence the set of divisors of a nonzero integer is bounded above by $|a|$).

Integer absolute value is defined by sign cases (The absolute value $|a|$ of an integer).

Integer absolute value is nonnegative and detects zero (Absolute value in $\mathbb{Z}$: $|a| \ge 0$; $|a| = 0$ exactly when $a = 0$; $|-a| = |a|$; $|ab| = |a|\,|b|$; $-|a| \le a \le |a|$; and $|a| \le c$ exactly when $-c \le a \le c$).

The integers form an ordered ring (The integers form a totally ordered ring).

Natural numbers embed in the integers preserving arithmetic (The naturals embed in the integers).

The natural order is discrete (Discreteness: $\sigma(n)$ is the immediate successor).

For prime $p$, the ring $\mathbb Z/p$ is a field (For every prime $p$, the two operations on $\mathbb{Z}/p$ make it a field).