Talk:Abstract Algebra/Number Theory

Nice Job
I saw you wrote more. That is nice. I am at the group section. Are you interested? --Arydye001 (talk) 00:41, 19 July 2010 (UTC)

Scope of this page?
I was going to review this revision, but I can't rate the coverage as I don't know how much of number theory we are intending to cover within this book / on this page. Obviously the entire field of number theory can't be covered, but do people have an idea what would be a good range to aim at for an introductory chapter? --Fishpi (talk) 09:32, 18 July 2010 (UTC)
 * I have seen abstract algebra textbooks cover number theory in great detail, and many of the concepts introduced in number theory can be useful in abstract algebra. Consequently, I (who authored this page but am unsure of the original intention of authors of other portions of the book) thought to attempt to cover definitions of the natural numbers and integers, then move to divisibility and factorization, and then discuss modular arithmetic in some detail. Would less coverage perhaps be more appropriate? --Vorziblix (talk) 01:36, 19 July 2010 (UTC)


 * The most important question to me is what the minimum requirements are for an abstract algebra book, so we can mark the page as quality-reviewed once it meets these requirements. I certainly don't object to having more than the minimum amount of information here.
 * It seems to me that the parts of number theory most relevant to abstract algebra are things like prime factorisation, modular arithmetic etc. I wonder if it would make more sense to start from an intuitive approach to number theory rather than the Peano axioms? I feel like the axiomatic approach might be too cumbersome in covering all the material necessary. --Fishpi (talk) 10:53, 19 July 2010 (UTC)


 * Yes, I would agree that the areas you mentioned are the most relevant, and they should certainly be included in any minimum requirements. As for the axiomatic and intuitive approaches, I thought to begin with an axiomatized system to provide a firm foundation for the field and then transition to more intuitive approaches when the natural numbers and integers had been defined. Perhaps an intuitive introduction should be included alongside the deductions from axioms? --129.138.44.92 (talk) 06:09, 20 July 2010 (UTC) --Vorziblix (talk) 06:10, 20 July 2010 (UTC)


 * I've not seen a number theory text book that starts with the Peano axioms. I don't think most people think in axiomatic terms when learning number theory. Perhaps we could refer readers to a mathematical logic wikibook (not that there is one yet) if they want to know more about the axiomatic basis? Anyway, since you're contributing to this book and I'm not, I'm happy to let you go ahead and set it out how you want. --Fishpi (talk) 07:17, 20 July 2010 (UTC)

Renaming this section?
Maybe it would be appropriate to rename this page to "Axiomatic foundation of the integers" or something, and have a separate page that covers modular arithmetic, gcd and lcm, prime factorization and the division algorithm? Espen180 (discuss • contribs) 10:09, 16 April 2012 (UTC)

Integer Section seems, wrong
First of $$\mathbb{Z}$$ does not represent the rational integers (whatever that means.) I think they mean the rational numbers $$\mathbb{Q}$$. But the axioms are essentially reversed for any kind of number.

TheKing44 (discuss • contribs) 14:32, 29 March 2013 (UTC)

They mean the integers. The construction given represents the integer z as the set of all natural numbers (n,m) such that n - m = z.

69.86.101.23 (discuss) 18:40, 8 October 2016 (UTC)

Notation
I noticed that the infix notation for addition is explicitly defined. In contrast, multiplication uses juxtaposition, but this isn't made explicit. I think it should be. Also, I think it would be easier to read if the page used a visible symbol for the multiplication operator (preferably *, which is standard) instead of juxtaposition. I also think it would be nice if the successor of x was written as S(x), because that would make the notation used here harmonize with the Wikipedia article on the Peano axioms. Thanks for reading! JonathanHopeThisIsUnique (discuss • contribs) 06:02, 28 May 2020 (UTC)