Simplify and extend introduction in ANT1#3
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| \noindent This construction is very powerful, and can be used to solve many classical problems, such as: | ||
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| \item \textbf{Diophantine Equations}: For example, solving the Diophantine equation $x^2 - dy^2 = 1$ (Pell's equation) can be understood as a problem in $\Z(\sqrt{d})$. Without going into detail for now, norm of $x + y\sqrt{d}$ in such ring is defined as $x^2 - dy^2$, which corresponds to equation we're interested in. |
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Should I nitpick that if d ≡ 1 (mod 4) we're actually working in ℤ[1/2(1 + √d)]? This doesn't sound too important in the motivational segment.
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It's not clear what Z(.) means. I'd stick with writing Z[.]. You can add a footnote noting that we consider Z[1/2(1 + √d)] when d ≡ 1 (mod 4).
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That's actually me being sloppy and not proofreading enough. I wasn't trying to use a clever notation. Thanks for spotting (I've removed this part entirely for now, though).
| \item The minimal polynomial of $1$ is $x-1$. So $\deg 1 = 1$. | ||
| \item The minimal polynomial of $\frac{1}{2}$ is $x - \frac{1}{2}$. So $\deg \frac{1}{2} = 1$. | ||
| \item The minimal polynomial of $\sqrt{2}$ is $x^2 - 2$. So $\deg \sqrt{2} = 2$. | ||
| \item The minimal polynomial of $i$ is $x^2 + 1$. So $\deg i = 2$. | ||
| \item The minimal polynomial of $\sqrt[3]{2}$ is $x^3 - 2$. So $\deg \sqrt[3]{2} = 3$. |
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These are the long and frequent examples I've mentioned in the description. I may be overdoing this here with 5 examples of minimal polynomials, but on the other hand I don't see how it can hurt.
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Hi! Sorry for pinging. Do you think this pull request is a good fit for this project? if not - for example you think it's too verbose, or you're not happy with quality, please let me know. Also, I know maintaining a project is a lot of work, and I understand if you don't have time - or don't want to - check my latex. If I can help somehow (for example, I can attach a pre-rendered PDF here), let me know too. In case this gets merged, I thought about slowly working on ANT chapters (or rather sections) and making them (hopefully) clearer, more approachable, and more motivated with examples and applications. |
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I'm currently busy so won't get to it until this weekend. I do appreciate your contributions! |
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The reason these terms haven't been defined is that they're intended to be a introduced in a previous "book" Field and Galois Theory (fgt), which gives the necessary background. (I intended to have the material from an abstract algebra course, e.g., Artin's Algebra, be in the fgt chapters.) Those chapters are however incomplete. |
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More precisely, I would put the background on fields and algebraic numbers in a separate chapter (you can check the fgt chapters for existing material). |
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Thanks for your review! Yes, I see your point. To be honest I didn't want to write a new chapter, but I started working on the "integrality" section and - when trying to define everything that's used - I apparently wrote enough to create a short chapter. My current plan is to continue working on the algebraic number theory segment for a while. Actually I have a three pull requests for the "rings of integers" chapter lined up, but I don't want to overwhelm you by submitting them all at once. Long term, if I manage to keep contributing, I will move the algebraic segment to the FGT book. And maybe expand that book a little? We'll see. Attached is a PDF file with just the changed pages, to (hopefully) make review easier. By the way, originally when submitting this PR I didn't notice there's over 100 pages on Field/Galois theory and elementary number theory. I realised this a day or two later when searching for things in the latex source. I think this project does itself a disservice by hiding this so much, and readers could benefit from having more content, even if in a rough state. But that's another topic entirely. |
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| \chapter{Rings of integers}\llabel{ring-of-integers} | |||
| When we have a field extension $L$ of $\Q$, we would like to define a ring of integers for $L$, with properties similar to the ring $\Z\subeq \Q$. We will define this ring of integers in a slightly more general context. | |||
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No more long introduction in this chapter so no reason to remove this now, but I plan to send some changes to the integrality section next, and I changed this sentence anyway.
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By the way: I understand that you're busy, and there's no rush in replying and reviewing. My impatience before was because I was still uncertain if you're even interested in my changes and I wanted to get a clear signal. |
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This PR is inspired by me first opening the book and being hit by a pretty dense math from the first page. I know some basic ANT (and I'm actively learning), so I figured I'll try to ease potential readers into the book.
In particular, I think my contribution fits the guidelines as outlined in the README:
I've added a "motivation" section at the beginning of the chapter to explain what rings of integers are useful for.
Similarly, "motivation" section should hopefully explain why the material matters. I've also tried to ease the reader into the topic of ANT with algebraic numbers, before the rest of the chapter generalizes this to arbitrary fields.
In my introductory part I've defined multiple terms used previously without definition, including minimal polynomials, conjugates and number fields
Nevertheless, as mentioned before, I'm very far from being an expert on the subject (and LaTeX), so this PR may need a careful review.
Additionally, my style stands out from the rest of the book a little - I added significantly more concrete examples than any other place in the book. If that's not welcome, they may be removed (or reduced), though I consider them educational and useful for learning.