What is the definition for totally ramified extension for a global field? For local fields it means the maximal prime ideal generated from the uniformizer totally ramifies. But what is the definition for an algebraic number field?
2026-03-25 22:04:31.1774476271
What is the definition for totally ramified extension for a global field?
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If $K$ is a global field, $p$ is a prime of $\mathcal{O}_K$, and $L/K$ is an extension with $n=[L:K]$, we say that $p$ is totally ramified in $L$ (or that $L$ is totally ramified at $p$) if $p\mathcal{O}_L=q^n$ for some prime $q$ of $L$.
However, since of course there are many primes of $K$, it's not a complete statement to only say "The field $L$ is totally ramified over $K$", you have to specify which prime you're talking about – alternatively, the statement could just mean "There exists at least one prime $p$ of $K$ that is totally ramified in $L$".