Proof of no rational point on Selmer's Curve $3x^3+4y^3+5z^3=0$ 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?
 A: This problem is in Cassels' book "Local Fields" and I wrote up a solution once along those lines, for an algebraic number theory class.  See 
http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/selmerexample.pdf,
but I should advise that it comes out seeming pretty tedious.  Solutions that involve elliptic curves are more conceptual.  Others have already provided pointers to references for that approach.
A: 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.
A: I think I saw a proof of that in Cassel's "Diophantine Equations with special reference to elliptic curves" and in some surveys by Mazur in the Bull. AMS (perhaps this, but I have in the moment no time to look).
A: There's a proof in Cassels' little blue book on elliptic curves which the OP might find more to his taste than some others mentioned here.
A: The "standard" technique for killing the Hasse priniciple for elliptic curves is to show that the Tate-Shafarevich group has a copy of (Z/mZ)^2 for some m - see chapter X in Silverman's the arithmetic of Eliptic curves, both for the theory and examples. All the examples which Silverman presents ar with m = 2. Selmers example requires m = 3, which requires (much) more computations. Poonen has an example
on his web page of a family of elliptic curves violating the Hasse principle, and containing Selmers example, but you'd have to dive through a labirinth of references.
