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Let us consider the $p$-adic numbers $\mathbb Q_p$ and the consider the tower of Galois extensions: $$\mathbb{Q}_p \subset K \subset L; ~[L: \mathbb Q_p]<\infty$$ where

  • $K$ is an unramified extension of $\mathbb{Q}_p$.

  • $L$ is a totally ramified extension of $K$.

If $M$ is a maximal unramified extension of $K$ inside $L$, then what can we say about $M$?

My calculation says, $M$ is just equal to $K$, that trivial one. Because, $L$ being totally ramified extension of $K$, there is no extension of their residue fields, that is, $\kappa_K=\kappa_L$. Since $M$ lies inside $L$, its residue field can not be larger than $\kappa_L$ and which is equal to $\kappa_L$. So $M=K$.

Am I missing anything?

Learner
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    this is essentially just a rephrasing of what you've said, but maybe is useful "shorthand": ramification $e$ and inertia $f$ are multiplicative in towers of extensions, so $f = 1$ for any intermediate extension by the assumption on $L$. $[M : K] = ef$ and $e = 1$ by assumption so $M = K$. – user1090793 Jun 24 '25 at 20:01
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    So, it is true for any unramified extension of $K$ inside $L$, not necessarily the "maximal" unramified one; in other words, $L$ being a totally ramified extension of $K$ implies that $K$ is the maximal unramified extension of $\mathbf{Q}_p$ inside $L$. – Anuradha N. Jun 25 '25 at 19:50

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