3.II.19F

Explain what is meant by saying that a positive definite integral quadratic form $f(x, y)=a x^{2}+b x y+c y^{2}$ is reduced, and show that every positive definite form is equivalent to a reduced form

State a criterion for a prime number $p$ to be representable by some form of discriminant $d$, and deduce that $p$ is representable by a form of discriminant $-32$ if and only if $p \equiv 1,2$ or $3(\bmod 8)$. Find the reduced forms of discriminant $-32$, and hence or otherwise show that a prime $p$ is representable by the form $3 x^{2}+2 x y+3 y^{2}$ if and only if $p \equiv 3(\bmod 8)$.

[Standard results on when $-1$ and 2 are squares (mod $p$ ) may be assumed.]

*Typos? Please submit corrections to this page on GitHub.*