Let p ≡ 1 (mod 3) be prime. Show that there exists an integer a such that a^2 + a + 1 ≡ 0 (mod p).
Then show that p = x^2 + xy + y^2 for some integers x and y.
In all honesty I have very little idea as to where to begin with this problem. Having completed several proofs (which appear similar) which rely on both quadratic reciprocity and the use of Minkowski's Theorem, I have attempted to use these techinques here, though to little success.
If anybody could help to get me started then that would be very much appreciated! Many thanks!
This is just an idle thought, but I wonder if you could get somewhere with this part of the problem by mimicking Lagrange's proof of Fermat's theorem about sums of two squares?
Originally Posted by Rocky
I have a thought, but I don't know much about quadratic residues. Perhaps someone can pick up where I leave off.
The discriminant must be equal to . Does quadratic residues have something to say about this one?