Let $K$ be a local field and $K'$ be it's finite extension. Then, why ${K'}^{nr}/K^{nr}$ is finite extension ?
Unramified extension of local field corresponds to extension of residue field. So, I can translate this into the problem of finite field. Let $k$ be a finite field, and $k'$ be it's finite extension. Then, why algebraic closure of $k'$ is finite extension of $k$ ?
This is related to the Silverman's book 'the arithmetic of elliptic curves', $p202$, $Cor7.3$.
Thank you in advance.