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](http://en.wikipedia.org/wiki/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?



Proof of no rational point on Selmer's Curve $3x^3+4y^3+5z^3=0$
My friend has written an introduction to algebraic number theory before, which contains a short proof of this statement, but I didn't check its validity.


**Edit:** updated the link of the document, <http://www.2shared.com/document/2d6M7kNU/Introduction_to_Algebraic_Numb.html>
p. 41 of the document, or p. 45 of the PDF.


