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I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theorem:

$(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value extension, $L/K$ a finite separable field extension.

Then the $K_u$-algebra $K_u\otimes_K L \approx \prod_i (K_u\otimes_K L)\eta_i \approx \prod_{w_i} L_{w_{i}}$.

( $\eta_i$'s are idempotents of $K_u\otimes_K L$, $L_{w_i}$'s are finite separable field extension of $K_u$ where $w_i$ runs through all absolute values of L which extend u.)

I have no problem with this theorem, my question is the next proposition:

With the same notation above, let $\mathfrak p$ be a prime ideal of a Dedekind domain $\mathfrak o$ with field of fraction $K$, and if $u$ is a discrete absolute value of $K$ corresponding to $\mathfrak p$, then, writing $\mathfrak o_{K_u}$ for the valuation ting in $K_u$($\mathfrak o_{K_u}=\{x;\bar u(x)\leqslant 1\}$), we have $\mathfrak o_{K_u}\otimes_{\mathfrak o_K} \mathfrak o_L \approx \prod_i (\mathfrak o_{K_u}\otimes_{\mathfrak o_K} \mathfrak o_L)\eta_i \approx \prod_{w_i} \mathfrak o_{L_{w_i}}$

The book doesn't specify what $\mathfrak o_L, \mathfrak o_K, \mathfrak o_{L_{w_i}}$ are, but with the convention notations in the book, it might mean valuation rings in $L, K, L_{w_i}$ respectively.

However, since $L$ has multiple distinct absolute values ($w_i$'s), it's not clear $\mathfrak o_L$ corresponds to which absolute value of $L$, hence I have no idea what this property is talking about. What are $\mathfrak o_L, \mathfrak o_{w_i}$?

This is its proof which may help understanding: enter image description here

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  • $\begingroup$ Please note that "theory" and "theorem" are different words. $\endgroup$ – Zev Chonoles Aug 23 '15 at 14:39
  • $\begingroup$ @ZevChonoles Thanks, I am not naive English speaker and sometimes misuse. $\endgroup$ – CYC Aug 23 '15 at 14:44
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I would say from context that $\mathfrak{o}_L$ and $\mathfrak{o}_K=\mathfrak{o}$ are the rings of integers in $L$ and $K$ respectively (thus, the intersection of all the valuation rings for the different non-archimedean valuations) and $\mathfrak{o}_{L_{w_i}}$ is the valuation ring for $w_i$.

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The notation $\mathfrak{o}_K$ (hence also $\mathfrak{o}_L$) is explained on Page 2 of the book:

The most important such subring is $\mathfrak{o}_K$, the set of all algebraic integers in $K$: later we shall show that $\mathfrak{o}_K$ is indeed a ring.

The notation $\mathfrak{o}_{L_{w_i}}$ should be clear from the context, since it is analogous to $\mathfrak{o}_{K_u}$.

Finally, a word of advice that also applies to your previous question: if you quote from a book please do it precisely (word by word), and also give the exact page/section number.

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