The polynomial $x^2-2$ is irreducible over $\mathbb Q$

Given: The polynomial $f=x^2-2\in\mathbb Z[x]\subseteq\mathbb Q[x]$.

A reduced rational root $r/s$ of an integer polynomial has numerator dividing the constant coefficient and denominator dividing the leading coefficient (Rational root theorem).

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

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 $p$).

A divisor of a nonzero integer has absolute value at most the absolute value of that integer (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 given by the positive and negative cases (The absolute value $|a|$ of an integer).

Integer absolute value is nonnegative and vanishes exactly at 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).

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

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

The rational numbers form a field (The rationals form a field).