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Dec 18, 2013 at 0:48 comment added KConrad Mariano: the page you are probably thinking of is mathoverflow.net/questions/10535/…. Two answers there mention a proof by the Lefschetz fixed point theorem that is essentially the proof you write above.
Dec 18, 2013 at 0:17 comment added Mariano Suárez-Álvarez This is also a variation of the proof linked to by David in the comments to the question, of course.
Dec 18, 2013 at 0:12 comment added Mariano Suárez-Álvarez Somewhere on MO there is a sketch of a proof of the FTA which as above turns the projective space $P(K)$ into a Lie group, observes it is commutative, invokes the classification of compact abelian Lie groups and using, say, the computation of the cohomology of tori and projective spaces, notes that no positive-dimensional torus is a projective space. My argument above is a minor variation, really.
Dec 18, 2013 at 0:07 history answered Mariano Suárez-Álvarez CC BY-SA 3.0