The Stacks project

Remark 111.14.2. A singularity on a curve over a field $k$ is called an ordinary double point if the complete local ring of the curve at the point is of the form $k'[[x, y]]/(f)$, where (a) $k'$ is a finite separable extension of $k$, (b) the initial term of $f$ has degree two, i.e., it looks like $q = ax^2 + bxy + cy^2$ for some $a, b, c\in k'$ not all zero, and (c) $q$ is a nondegenerate quadratic form over $k'$ (in char 2 this means that $b$ is not zero). In general there is one isomorphism class of such rings for each isomorphism class of pairs $(k', q)$.