Let $L/K$ be a finite unramified extension of non-Archimedean local fields with rings of integers $\mathcal O_L$ and $\mathcal O_K$, maximal ideals $\mathfrak m_L$ and $\mathfrak m_K$, and residue fields $\ell=\mathcal O_L/\mathfrak m_L$ and $k=\mathcal{O}_K/\mathfrak{m}_K$, then a paper I read assumed that $\ell$ must be a simple extension of $k$, i.e. there exists some $\bar{x}\in\ell$ such that $k(\bar{x})=\ell$, and so I presume that it is the case that $\ell/k$ is separable.
But why is this true?