Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 20H / a / i / Solution

The trace form of a number field isIt is bilinear because the field trace is linear. If $x\ne 0$, choose $y=x^{-1}$. ThenThus no nonzero $x$ lies in the radical, and $B$ is a nondegenerate bilinear form. This is the nondegeneracy of the trace form of a number field.

Solved by gpt-5.6-sol high.