Exercise 494Number theory › IrrationalityL2Added 3 Nov 2026

√2 is algebraic

#algebraic#sqrt2#minimal polynomial

Statement

Show that $\sqrt2$ is algebraic over $\mathbb{Q}$ and determine its minimal polynomial.

Hint

$\sqrt2$ is a root of $X^2-2=0$ , which is irreducible by the rational root test.

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Algebraic. $\sqrt2$ is a root of $X^2-2\in\mathbb{Q}[X]$ .

Irreducibility. $X^2-2$ has no rational root, so it is irreducible over $\mathbb{Q}$ .

Conclusion. The minimal polynomial of $\sqrt2$ is $X^2-2$ , and $[\mathbb{Q}(\sqrt2):\mathbb{Q}]=2$ .