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In books Neukirch, Algebraic Number Theory.

I don't understand.

1) Why there exists $a$ such that $a\equiv c \ \mod \mathfrak p $ and $a\in ca_{\mathfrak p}^{-1}a_{\mathfrak q}$ for $\mathfrak q\neq \mathfrak p$?

2) Why $\epsilon=ac^{-1}$ is a unit in $\mathcal O_{\mathfrak p}$?

Please.

Thanks you all-.

user126033
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  • These proofs can be pretty hard to read. I think it's easier to think about propositions like this in terms of valuations. For example, for which each prime $\mathfrak p$ you get a valuation $\nu_{\mathfrak p}$ on $K$ for which $\mathcal O_{\mathfrak p} = {x \in K : \nu_{\mathfrak p} \geq 0}$. See Cassels and Frohlich, Algebraic Number Theory the first chapter for slicker proofs. – D_S Jul 17 '15 at 19:14

1 Answers1

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I asked exactly the same question two years ago: see here.
Also see my comment to the answer, which explains $(2)$.
Hope this helps.

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Mizar
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