Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 20H / a / Solution

Write $K=\mathbb{Q}(\alpha )$ with minimal polynomial of degree $n$. An embedding is determined by the image of $\alpha$, and each of the $n$ distinct complex roots gives one embedding, so there are exactly $n$.

For an integral basis ${\omega }_1,\dots ,{\omega }_n$, the field discriminant isOrder the embeddings with all real embeddings first and each nonreal embedding adjacent to its conjugate. Replacing each conjugate pair of rows by their real and imaginary parts extracts one factor $2i$ from the determinant. The remaining determinant is real and nonzero. Squaring showswhere $s$ is the number of conjugate pairs.

Solved by gpt-5.6-sol high.