Talk:Binary quadratic form

Assessment comment

The comment(s) below were originally left at Talk:Binary quadratic form/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

The class number in binary quadratic forms should be same as the class number in the ideal classes of algebraic number fields, so the connection should be pointed out.Bergv (talk) 01:54, 19 October 2009 (UTC)

Last edited at 01:54, 19 October 2009 (UTC). Substituted at 01:48, 5 May 2016 (UTC)

Requested move 27 February 2017

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: not moved. Seems to be agreement that redefining the scope of the article in the lead is a better idea than renaming the article. Discussion about that can continue below. Jenks24 (talk) 14:44, 9 March 2017 (UTC)

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Binary quadratic form → Integral binary quadratic form – most people searching for "binary quadratic form" are actually looking for information about a specific type of binary quadratic form, namely, one whose coefficients are integers. I just went through all the links at the "what links here" page and found that most other pages explicitly state that the quadratic forms in question have integer coefficients, and most of the remaining ones do this implicitly. An exception is the page Paul Bernays -- he did in his work consider binary quadratic forms with more general real number coefficients. Standard textbooks, such as "Binary Quadratic Forms" by Duncan Buell, consider forms with only integer coefficients. Recent redirect traffic to this page only came through "reduction theory of forms", "class number (binary quadratic forms)" and "composition of forms". Before I edited the page this week, the brief mentions of these topics here only concerned integral forms. In my research on binary quadratic forms, I have seen one paper out of dozens discuss reduction of non-integral forms, and I have never seen any mention of composition of forms that weren't integral. Planetmath.org titles their main page for quadratic forms "integral binary quadratic forms" (although that page has some broken functionality). If the move I am suggesting seems sensible and is made, then "binary quadratic form" should redirect to this page and the hatnote should indicate that people searching for binary quadratic forms with other types of coefficients should see quadratic form. Barryriedsmith (talk) 23:35, 27 February 2017 (UTC)

• Weak oppose I agree that this article should be only about integral binary forms. In fact, there are very few results on binary quadratic forms with non-integer coefficients that are specific to the binary case. However the proposed title is very long, and would not be easily found by a direct search (without passing through a redirect). Therefore, I propose to keep the present title, and simply add the hatnote

This article is about binary forms over the integers. For for other binary forms, see Quadratic form.

Clearly a change of the lead would be needed, but it would also needed if the move is done. D.Lazard (talk) 15:00, 28 February 2017 (UTC)

This makes sense to me, and I even thought about just making this change. My issue came when thinking about how I would rewrite the lead. It seemed to me it would go something like "a binary quadratic form is a homogeneous polynomial in two variables $ax^{2}+bxy+cy^{2}$. When a, b, and c are integers, it is called an integral binary quadratic form. This page describes only integral binary quadratic forms..." That last sentence is redundant with the hatnote. But if we omit it, I feel we are assuming to much when we assume the reader has read and understood the hatnote before reading the lead. If we omit the hatnote, it still seems clunky to say "this page describes only integral binary quadratic forms" in the lead. Another approach would be to make the first sentence "a binary quadratic form is a homogenous polynomial in two variables $ax^{2}+bxy+cy^{2}$ with integer coefficients," but that seems to me at best misleading. Perhaps there is a standard way to write the lead for a page whose title is more general than the actual content? A way that is precise, not clunky, and doesn't rely on the reader observing and understanding the hatnote?Barryriedsmith (talk) 15:34, 28 February 2017 (UTC)

Why do not explain clearly the reasons of this unusual behaviour, with something like "In mathematics, a binary quadratic form is a homogeneous polynomial in two variables $ax^{2}+bxy+cy^{2}.$ When the coefficients a, b, c are not necessarily integers, most results are not specific the the case of two variables, and, therefore, are described in Quadratic form. When the coefficients are integers, one talks of an integer binary quadratic form, often abbreviated to binary quadratic form, or even, simply, quadratic form. This article is entirely devoted to integer binary quadratic forms, and this is motivated by the importance of their applications to number theory."?

The fact of referring to number theory in the first paragraph is also useful for introducing a second paragraph devoted to a better description of the main content of the article. D.Lazard (talk) 16:16, 28 February 2017 (UTC)

That sounds good -- I like the reference to number theory where you put it for the reason you state. This lead would make the hatnote redundant, but I guess that's not a big deal, right? And the hatnote would still be a quicker way for some folks to find what they're looking for rather than reading through the whole lead.Barryriedsmith (talk) 16:48, 28 February 2017 (UTC)

I agree that the article should mostly be about quadratic forms with integer (or rational) coefficients. On the other hand, by "integer" one usually means a rational integer (element of $\mathbb {Z}$) as opposed to an algebraic integer (element of the integral closure of $\mathbb {Z}$ in $\mathbb {C}$). There is a substantial litterature on binary quadratic forms with algebraic integral coefficients and while it is not mentioned in the article as it stands I am not sure that preventively restricting the article to "integral quadratic forms" in the lead is the best idea. So I would replace entirely in D. Lazard's proposal above by mostly (though of course these more general forms are still connected to number theory, and another more different formulation might be preferable). Regards, jraimbau (talk) 12:51, 1 March 2017 (UTC)

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The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.