Coprime-degree descent for powers
= Coprime-degree descent for powers
Let $L/K$ have degree $d$ with $\gcd(d,n)=1$. If $a=x^n\in K$ for some $x\in L$, then
$$
a^d=N_{L/K}(x)^n.
$$
Choosing $r,s\in\mathbb Z$ with $rd+sn=1$ gives
$$
a=\bigl(N_{L/K}(x)^r a^s\bigr)^n,
$$
so $a$ was already an $n$th power in $K$.