= Unramified extensions remain unramified under base change
{title2=$L/K\text{ unramified}\ \Longrightarrow\ LE/E\text{ unramified}$}
For non-Archimedean <local fields>, let $L/K$ be an <unramified extension> of degree $f$, and let $E/K$ be finite. The degree-$f$ <unramified extension> of $E$ has a <residue field> containing the degree-$f$ residue extension of $K$. Lift an integral residue generator of $L$ into that field by <Hensel lemma>. This embeds $L$ into an <unramified extension> of $E$, so $LE/E$ is an <unramified> subextension. Applying this argument to each <completion of a number field at a prime ideal> proves the finite-prime assertion for <number fields>, even when $E/K$ is ramified.
Back to article page
