Norm form

For a finite number field $L/\mathbb{Q}$ with a rational basis ${\omega }_1,\dots ,{\omega }_n$, the norm form is the homogeneous polynomial

For a rational prime $p\ne 229$, representation by this norm form is equivalent to complete splitting in the Hilbert class field of Q of square root 229. Its Frobenius is a permutation of the three roots of $X^3-4X-1$. The quadratic residue condition $(p/229)=1$ makes this permutation even; a root modulo $p$ gives a fixed point, so the even permutation is the identity. Complete splitting makes the prime above $p$ principal. Its generator has norm $\pm p$, and the unit of norm minus one changes the sign to positive, giving the representation.