Proof of no rational point on Selmer's Curve $3x^3+4y^3+5z^3=0$

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.

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.

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).