To prove the quadratic reciprocity law, Gauss needed the following lemma:
If $p$ is a prime number congruent to 1 modulo 8, then there exists a prime $q<p$ such that $p$ is a non residue modulo $q$.
Gauss demonstrated this result in no 126, 127, 128, 129 of the Disquisitiones, and it is in fact the essential difficulty of the quadratic reciprocity law (with this lemma, I could provide a terrible simplification of the original proof of Gauss by induction).
I found the demonstration of Gauss both beautiful and amazing, in some sense natural too. But I feel there is something more, hidden there, a general principle or a beautiful elementary lemma of modular theory (akin to Thue's lemma for example), which would provide a much more luminous and simpler demonstration to this result.
Can someone try to extract it? Or maybe someone already knows how to do that?
N.B: I'm not sure there is an available translation to English, but here is an available translation of the Disquisitiones in French (I know there is also one in German).