Is π √2 rational or irrational? And how would I prove this?

[lostinmath]

is π √2 rational or irrational and how do i prove this ?

 

[stapel]

is π √2rational or irrational and how do i prove this ?

What are your thoughts? What have you tried? Where are you stuck?

For instance, you know that you cannot say that any product of irrationals is irrational, since:

. . . . .\(\displaystyle \dfrac{1}{\sqrt{\strut 2\,}}\, \cdot\, \sqrt{\strut 2\,}\, =\, 1\)

...which is a rational result for a product of irrationals. You can't even say that a product of two transcendental irrationals is irrational, since you can replace "2" with "pi" and get the same result. So you already know, going in, that you may need to be using something specific to the square root of two and pi. If you've studied algebraic fields, comments such as this one may prove helpful. And so forth.

So what have you tried? Please be complete. Thank you!