Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 20G / a

Let ${\sigma }_1,\dots ,{\sigma }_n:K\hookrightarrow \mathbb{C}$ be the embeddings of the number field $K$. The discriminant of elements of a number field isIf the ${\alpha }_j$ are $\mathbb{Q}$-linearly dependent, the embedding matrix has dependent columns and the discriminant vanishes. Conversely, if they are independent, they form a $\mathbb{Q}$-basis of the $n$-dimensional vector space $K$. The trace pairing of a characteristic-zero number field is nondegenerate, so its Gram matrix in this basis is nonsingular. Hence