Is $\sqrt{2}$ a rationnal number ?

\sqrt{2}

a rationnal number ?

by Sequoia·

Let us assume that $\sqrt{2}$ is a rationnal number. Set $a,b$ two coprime integers such that $\sqrt{2}=\frac{a}{b}$ .

Show that this statement is absurd.

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Solution by Sequoia

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Squaring the equation gives $2=\frac{a^2}{b^2}$ and taking the $2$ -adic valuation leads to:
$1=v_2(2)=v_2\left(\left(\frac{a}{b}\right)^2\right)=2\,v_2\left(\frac{a}{b}\right)$ But $v_2\left(\frac{a}{b}\right)$ is an integer and $1$ is not an even integer so this is absurd.

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