The splitting field of $x^2-2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt{2})$, with roots $\pm \sqrt{2}$

Given: The polynomial $x^2-2\in \mathbb{Q}[x]$ and the positive real square root $\sqrt{2}$.

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

The nonnegative real number $2$ has a unique nonnegative square root $\sqrt{2}$ with $(\sqrt{2}{)}^2=2$ (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$).

Eisenstein's criterion proves a primitive integer polynomial irreducible when a prime divides every nonleading coefficient, its square does not divide the constant coefficient, and it does not divide the leading coefficient (Eisenstein criterion over the integers).

A simple extension by an algebraic element whose minimal polynomial has degree $n$ has power basis through degree $n-1$ (A simple algebraic extension is its minimal-polynomial quotient and has power basis $1,a,\dots ,a^{n-1}$ and degree $n$).

A splitting field is generated by all roots of the polynomial (Polynomials that split and splitting fields of a polynomial or a family of polynomials).

The minimal polynomial of an algebraic element is the unique monic irreducible polynomial that vanishes at it (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element).