= Quaternion norm obstruction at primes congruent to three modulo four
{title2=$v_p(N_{\mathbb Q(i)/\mathbb Q}z)\in2\mathbb Z$}
For $p\equiv3\pmod4$ and $z=x+iy\ne0$ with rational $x,y$, divide $x^2+y^2$ by the square of the common minimum $p$-adic valuation. Its remaining unit cannot vanish modulo $p$, because $-1$ is not a square there. The norm therefore has even valuation. In particular a rational parameter $a$ with odd valuation at such a prime is not a norm, and $(-1,a)_{\mathbb Q}$ is a division <quaternion algebra>: its norm $N(z)-aN(w)$ cannot vanish for a nonzero element $z+w j$.
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