Isotropy of nondegenerate ternary quadratic forms over finite fields
= Isotropy of nondegenerate ternary quadratic forms over finite fields
Over a <finite field> of odd cardinality $q$, a diagonal nondegenerate <ternary quadratic form> $A x^2+B y^2+C z^2$ has a zero with $z=1$: the sets $\{A x^2\}$ and $\{-C-B y^2\}$ both have $(q+1)/2$ elements and must intersect. This gives a nonzero nonsingular zero. Over a local integer ring with these coefficients units, the <Hensel lemma> lifts it to an isotropic vector in the fraction field.