= Prime representation by the norm form of Q of square root 229
{title2=$p=x^2+xy-57y^2$}
For a rational prime $p\ne229$, 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.
Back to article page
