Field elements which become $n$th powers after adjoining an $n$th root of unity

Question

Let $n$ be a natural number, $K$ a field of characteristic not dividing $n$. Let $L/K$ be the field extension of $K$ obtained by adjoining a primitive $n$th root of unity.

Can there be elements of $K$ which are an $n$th power in $L$, but not in $K$?

If $n$ is prime, then the statement follows by observing that if $a \in K$ such that $a$ is not an $n$th power in $K$, then adjoining an $n$th root of $a$ defines a degree $n$ extension, which cannot be contained in $L$, which is of degree dividing $n-1$.

What if e.g. $n=4$?

Answer 1

$-4=(1+i)^4$ became a fourth power when $i$, a primitive fourth root of unity, was adjoined to $\Bbb{Q}$.

Answer 2

Take $K=\mathbb R$ and $L=\mathbb C=K(i)$. Then all elements of $K$ become fourth powers in $L$. Note that $i$ is a primitive fourth root of unity. Only the positive real numbers are already fourth powers in $K$.