$K(u, v) = K(u+v)$ when characteristic of $K=0$?

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Let $F$ be a field extension of $K$ where $u, v$ are in $F$ such that $u$ is separable over $K$ and $v$ is purely inseparable over $K$. The answer to this question shows that $K(u, v) = K(u+v)$ when characteristic of $K = p > 0$. Does this still hold when the characteristic of $K=0$?