Local field extension and its residue extension

Question

If an extension $K/L$ of nonarchimedean local fields is finite Galois, then we have an extension of their residue fields $\kappa/\lambda$. Then we can define the extension to be unramified if their degrees coincide. Do we always have an extension of residue fields $\kappa/\lambda$ for any (finite) extension of nonarchimedean local fields?

Answer 1

If $K$ is an extension of $L$, then the rings of integers $\mathcal O_K$ is an extension of $\mathcal O_L$, so we have an embedding $\lambda = \mathcal O_L/\mathfrak m_L \to \mathcal O_K/\mathfrak m_K = \kappa$.