The projective curve $3x^3+4y^3+5z^3=0$ is often cited as an example (given by Selmer) of a **failure of the Hasse Principle**: the equation has solutions in any completion of the rationals $\mathbb Q$, but not in $\mathbb Q$ itself.

I don't think I've ever seen a proof of the latter claim — is someone able to provide an outline? What are the necessary tools?