$\Bbb Q(\sqrt 2)$ and $\Bbb Q(\sqrt 3)$ are not isomorphic How to prove that $\Bbb Q(\sqrt 2)$ and $\Bbb Q(\sqrt 3)$ are not isomorphic. I thought that they are but I got this problem in Dummit Foote Section 14.1. Question no 4. As they extension over $\Bbb Q$ by the polynomials $x^2-2$ and $x^2-3$ resp.

Hint:
Take $\;w=a+b\sqrt2\in\Bbb Q(\sqrt2)\;$ s.t. $\;\phi w=\sqrt3\in\Bbb Q(\sqrt3)\;$ , with $\;\phi\;$ an isomorphism. This means that
$$3=\phi w^2=\phi(a^2+2b^2+2ab\sqrt2)=a^2+2b^2+2ab\phi\sqrt2\implies\phi\sqrt2\in\Bbb Q$$
and now get a contradiction...
