= Solution
Put $d=[L:K]$. If $\alpha=0$ the result is immediate, so suppose $\alpha=x^n\ne0$ with $x\in L$. The <field norm> is multiplicative, sends an element of $K$ to its $d$th power, and commutes with powers. Therefore
$$
\alpha^d=N_{L/K}(\alpha)=N_{L/K}(x)^n.
$$
Choose integers $r,s$ with $rd+sn=1$. Then
$$
\alpha=(\alpha^d)^r(\alpha^n)^s
=\bigl(N_{L/K}(x)^r\alpha^s\bigr)^n.
$$
The expression in parentheses belongs to $K$, so $\alpha$ is an $n$th power in $K$. This is <coprime-degree descent for powers>.
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