Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 18F / a

Finite means $[L:K]<\infty$; algebraic means every element satisfies a nonzero polynomial over $K$. Linear dependence among $1,x,\dots ,x^n$ proves finite implies algebraic. The algebraic closure $\overline{\mathbb{Q}}/\mathbb{Q}$ is algebraic but infinite.