If we have a set $M$ of $2019$ real numbers such that for every even $a$, $b$ of numbers of $M$ it is verified that $a^2+b \sqrt2$ is a rational number. Show that for all $a$ of $M$, $a\sqrt2$ is a rational number.
Source: 2019 Argentina OMA Finals L3 p4
Tags: rational, algebra
If we have a set $M$ of $2019$ real numbers such that for every even $a$, $b$ of numbers of $M$ it is verified that $a^2+b \sqrt2$ is a rational number. Show that for all $a$ of $M$, $a\sqrt2$ is a rational number.