Talk:Quadratic integer

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In mathematics, quadratic integers are the integer solutions of the equations of the form:[edit]

Dear User:TakuyaMurata,

integer or integral, that is the question.

I believe that it should be integer: as the solutions should be integer numbers.

Integral is an adjective and means (among other things)"necessary to complete something, something that cannot be left out". But I do not believe that it is an adjective describing integers.

TomyDuby (talk) 12:33, 18 April 2009 (UTC)

Assuming you're a native English speaker, maybe I shouldn't argue more. But I still think "integral solution" (by this we mean solutions are integers) is correct as "integral" means something related to integers. In fact, I found some uses of this: [1] [2] [3] . This makes sense, since one rarely says "integer equation" as opposed to "integral equation". The context usually makes it clear that one doesn't mean an equation involving integrals. But if you insist, I'm not going to oppose the change. It's a minor issue, after all. -- Taku (talk) 23:28, 18 April 2009 (UTC)
Dear Taku,
Thanks for your comments. It seems to me that you are right: the quotations and even Wiktionary under http://en.wiktionary.org/wiki/integral says that as an adjective in "(mathematics) Of or relating to an integer."
Great discussion!!!
TomyDuby (talk) 00:16, 22 April 2009 (UTC)

Chapter 3, Class number not defined.[edit]

This term, though used in this section, has never been defined.

TomyDuby (talk) 03:31, 19 April 2009 (UTC)

Class number is the order of the ideal class group. This definition of course begs the question: what is ideal class group? The point I'm trying to get is that somehow you first need to know Dedekind domain to understand this kind of stuff. I don't know how much (commutative) ring theory should be covered here. -- Taku (talk) 21:49, 21 April 2009 (UTC)
I think that I resolved this issue by adding a link to class number where this term is defined. This stuff is way above my current knowledge. Again, thanks for your contribution.
TomyDuby (talk) 00:27, 22 April 2009 (UTC)

ring of integers in Q[\sqrt{-19}][edit]

the ring of integers in Q[\sqrt{-19}] is incorrectly identified as Z[\sqrt{-19}], but -19 is congruent to 1 mod 4. —Preceding unsigned comment added by 165.91.100.161 (talk) 16:10, 4 November 2009 (UTC)

indeed. I've changed it. RobHar (talk)

The "[when?]" in the last section[edit]

I think in the last sentence, by after a hundred years the original author meant a hundred years since the idea about the whole class group thing begins, but I found no reference. Could someone point me some good books on this topic? Tony Beta Lambda (talk) 03:39, 16 June 2013 (UTC)

Problems with the definition[edit]

The Definition section currently reads

Quadratic integers are solutions of equations of the form:
x2 + Bx + C = 0
for integers B and C.

The next sentence refers to a quantity D and then in parentheses states that D is a square-free integer. There's no definition of D, although it is clear that it is supposed to be the discriminant, in this. However, even granted that, there is no requirement for D to be square free. For example, √5 satisfies the equation with B=0, C=−5 and D=20 is not square-free. Even worse, we probably want to include rational integers in the class too, for which we might have D=0. Deltahedron (talk) 18:09, 19 April 2014 (UTC)

I would suggest the following.
Quadratic integers are solutions of equations of the form:
x2 + Bx + C = 0
for integers B and C. The discriminant and a quadratic integer is termed real if D>0 and imaginary if D<0. We call integers "compatible" if the discriminants differ by a square factor.
The set of all quadratic integers is not closed even under addition. But the set of quadratic integers with compatible D forms a ring, and it is these quadratic integer rings which are usually studied. Medieval Indian mathematicians had already discovered a multiplication of quadratic integers of D, which allows one to solve some cases of Pell's equation.
I invented the term "compatible" ad hoc. Is there a reliable source to quote for this approach, and for the historical remarks? Deltahedron (talk) 18:22, 19 April 2014 (UTC)

PID & D>0 => euclidean; examples for euclidean, but not norm-euclidean quadratic integer fields[edit]

Please add someone the references: https://www.researchgate.net/publication/268442914_On_Euclidean_rings_of_algebraic_integers, http://www.mast.queensu.ca/~murty/harper-murty.pdf and http://www.math.clemson.edu/~jimlb/CourseNotes/AbstractAlgebra/EuclideanNotNormEuclidean.pdf I don't know the standard way to do this here. 132.230.30.102 (talk) 14:52, 2 May 2016 (UTC)

Thanks for providing these references. The two first links that you have provided are not really useful for the article, but I have copied from the second one the references to Weinberger and Harper. For Clark, I have copied the reference from the first page of the article, and added the link. You could have done this yourself. D.Lazard (talk) 15:48, 2 May 2016 (UTC)