Trace form

Summary

The trace form of a finite extension $L/K$ is the symmetric bilinear form $(x, y) \mapsto \mathrm{Tr}_{L/K}(xy)$ on $L$, built from the field trace. It is nondegenerate exactly when $L/K$ is separable. Its Gram determinant in a basis is the discriminant of that basis. Over $\Q$ the trace form is a quadratic form whose signature counts the real embeddings of a number field.