Unramified base change preserves total ramification
= Unramified base change preserves total ramification
If $L/K$ is an <unramified extension> and $M/K$ is a <totally ramified extension> of non-Archimedean <local fields> inside a fixed <algebraic closure>, then $LM/L$ is totally ramified of degree $[M:K]$. A <uniformizer> of $M$ has an <Eisenstein polynomial> over $K$. Since a <uniformizer> of $K$ remains a <uniformizer> in $L$, that polynomial remains Eisenstein over $L$ and generates $LM$.