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In my understanding, for a number field $K$, one can identify $K^{\times}$ with a subring of $\mathbb{A}_K$ (the adele ring), and then define the idele class group as $$0 \longrightarrow K^{\times} \longrightarrow \mathbb{A}_{K}^{\times} \longrightarrow C_K \longrightarrow 0 ,$$ Are there any conditions on $K$ that will make this exact sequence split?

Thank you

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