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CBirkbeck committed Oct 23, 2023
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Expand Up @@ -444,7 +444,7 @@ \subsection{Some Ramification results}
\begin{definition}\label{def:rel_different}

Let $K, F$ be number fields with $F \subseteq K$. Let $A$ be an additive subgroup of $K$. Let \[A^{-1}=\{ \a \in K | \a A \in \OO_K\}\] and
\[A^* = \{ \a \in K | \Tr_{K/F} (\a A) \in \OO_F\}.\] The relative different $\frak{d}_{K/F}$ of $K/F$ is then defined as $((\OO_K)^*)^{-1}$ which one checks is an integral ideal in $\OO_K$. This is also the annihilator of $\Omega^1_{\OO_K/\OO_F}$ if we want to be fancy.
\[A^* = \{ \a \in K | \Tr_{K/F} (\a A) \in \OO_F\}.\] The relative different $\gothd_{K/F}$ of $K/F$ is then defined as $((\OO_K)^*)^{-1}$ which one checks is an integral ideal in $\OO_K$. This is also the annihilator of $\Omega^1_{\OO_K/\OO_F}$ if we want to be fancy.
\end{definition}


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