Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 18H / b / Solution

Put $d=[L:K]$. If $\alpha =0$ the result is immediate, so suppose $\alpha =x^n\ne 0$ with $x\in L$. The field norm is multiplicative, sends an element of $K$ to its $d$th power, and commutes with powers. ThereforeChoose integers $r,s$ with $rd+sn=1$. ThenThe expression in parentheses belongs to $K$, so $\alpha$ is an $n$th power in $K$. This is coprime-degree descent for powers.