Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 136 / 1 / a / ii

Let $n=[L:K]$ and write $g(X)={\prod }_{j=1}^n(X-{\alpha }_j)$. Lagrange interpolation, followed by summing over the conjugates, shows that the trace-dual of the power basis $1,\alpha ,\dots ,{\alpha }^{n-1}$ is contained in $g^{\prime }(\alpha {)}^{-1}{\mathcal{O}}_K[\alpha ]$ and has the same determinant. Hence

For $L=\mathbb{Q}(\sqrt{3})$, the ring of integers of a quadratic field is $\mathbb{Z}[\sqrt{3}]$. Taking $g=X^2-3$ givesFor $L=\mathbb{Q}(\sqrt{5})$, use $\alpha =(1+\sqrt{5})/2$ and $g=X^2-X-1$. Then