Trace form of a number field ($B(x,y)={\mathrm{Tr}}_{K/\mathbb{Q}}(xy)$)

The trace form of a number field $K$ is the symmetric bilinear form

The pairing uses the field trace of the product, rather than taking a trace of each factor separately.

For nonzero $x\in K$, choose $y=x^{-1}$. Thenso the trace form is nondegenerate.