How to prove $\sqrt{3}$ is a irrational number using direct proof

Question

I can prove this easily using proof of contradiction method but I am stuck at proving this using direct proof

Answer 1

$\sqrt3$ is a root of $X^2-3$ which is irreducible over $\mathbb{Q}$ (Eisenstein)

Answer 2

Hint: You can use the fact that if $f(x)$ is a monic polynomial with integer coefficients, then any rational root of $f(x)$ is necessarily an integer. Now, applying this to the polynomial $f(x) = x^{2} − 3$ you can conclude that $\sqrt{3}$ is irrational number.