Add proof that O_K is a free module, and small changes #236
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| Which makes $(x, y) \mapsto \TrK(x \cdot y)$ \emph{almost} an inner product | ||
| (see \Cref{ch:inner_product_spaces}), | ||
| except that it is not positive definite (for example, in $\CC$, we have $\Tr((1+i)^2) = 0$). | ||
| But having the matrix invertible suffices to do the following: |
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This "feels" a bit silly that we're proving it the "wrong" way (the book so throughoutly uses the basis-less ("Math 55a") approach; in fact before this pull request there's exactly one more position in the book that transposes a matrix instead of taking the dual of a linear transformation), but for some reason the normal proof (the bilinear form (x, y) ↦ Tr(xy) is nondegenerate so we're done) doesn't really sink in for me, and Napkin didn't define what is a nondegenerate bilinear form anyway.
| If the field is an imaginary quadratic field, visualizing the class of an ideal is really easy: | ||
| because multiplication by a complex number corresponds to a combination of a scaling and a | ||
| rotation (i.e. it preserves angles), two ideals belong to the same class if they are similar, | ||
| that is, you can overlap one onto the another using rotation and scaling. |
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Is this what "similar" normally mean? (Note that reflection is excluded. Although I haven't checked including reflection causes problem...)
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This looks nice, thanks. |
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Also: I'm pretty sure about the correctness of the last proof although I can't find it in any book or reference. Most books use the second proof (dual lattice), only Neukirch use the first proof (divide by determinant), and nothing uses the third proof (technically Oggier's note tries to do that but that proof was pointed out to be broken) |
--- the book's introduction.
(This proof seems worth seeing for me, but writing in such a way to convince other people of it is generally a difficult task to do, I think. If anything, I think I've been relying on the book's introduction alone to convince myself the content is interesting...)
(I learnt that sending a 5000-line-diff pull request is a sure way to make sure it never gets reviewed, so here we go. Actually I also got one more for open mapping theorem...)
Re "chain map": actually there's only one more location in the book that calls it "chain map", but it's shorter and commonly used in the literature anyway, so why not.