Talk:Gaussian integer

message: I am only an amateur mathematician, and I don't know the term for numbers of the form a + b \sqrt{-2} \ \,  (or more generally a + b \sqrt{-k} \ \, ) with integers a and b (and k), but I do know that they are significant. I suggest that these two steps be taken:
* Someone more knowledgeable than myself should write an article about this ring (or such rings) in case such articles don't already exist.
* A link to this article should be provided within the article about Gaussian integers, at the very least in the "See also" section.
Thanks! 188.169.229.30 (talk) 13:14, 3 January 2012 (UTC)
replies: text: That is a good suggestion, and the article already has links to the numbers you mentioned, Eisenstein integers and quadratic integers, in the "See also" section and elsewhere.  Perhaps the "See also" links need brief explanatory notes? —Mark Dominus (talk) 14:37, 3 January 2012 (UTC)
thread_title:  Other kinds of "complex integers" 

message: I tried to change "The Gaussian integers are a special case of the quadratic integers." I got it wrong.  Thank you David Eppstein for reverting.
However I still think the original version is badly put. It is a messy mixture of singular and plural. Surely we are not trying to say merely "Every Gaussian integer is a quadratic integer": we are trying to say something about the Gaussian integers as a structure. So I've had a second go at editing it, by saying, "The Gaussian integers form a commutative ring, being a particular case of a commutative ring of quadratic integers." I hope I've got it right this time, but if not, perhaps someone could reword it to say something clear about the Gaussian integers as a structure, rather than just reverting. Prim Ethics (talk) 15:12, 15 December 2013 (UTC)
thread_title:  Relation to quadratic integers 

message: I have only some mathematics training (graduate level electrical engineering), and I was perusing the Gaussian Moat article out of curiosity and that article had a link pointing here for the definition of Gaussian primes.  I see the definition, but do Gaussian primes have the property that you can't factor them?  For example, for X = a + bi, is there no set of Gaussian integers Yi such that X = ΠYi?  That isn't clear to me from this or the prime elements article, and I think spelling it out would make the Gaussian prime section more accessible. - Daniel Morgret  — Preceding unsigned comment added by 74.96.218.225 (talk) 21:55, 26 December 2016 (UTC)
replies: text: I is always true that a prime element is also an irreducible element in an integral domain. The converse is true for unique factorisation domains, and thus for Gaussian integers. I have added this property to the article, and I hope that this answers your question. D.Lazard (talk) 11:37, 28 December 2016 (UTC)
thread_title:  Gaussian Primes 

message: has recently added a new section "Congruences and residue classes" to this article. This was lacking. However, this new section was written in a old fashioned style, without any reference to modern terminology. In particular, there was no mention that the congruence classes form the quotient of the Gaussian integer by an ideal, and no link to Quotient ring. Also, as it is heavily used that an ideal is a sublattice of the lattice of the Gaussian integer, there was no mention of that. I have thus rewritten this section in a more encyclopedic style (for an encyclopedia, links to related notions are fundamental).
 has just restored his version with the edit summary: . I am not willing to discuss the aggressive style of this edit summary (I have already recalled WP:Civility to this user, see above in this talk page). I want just to remark that a text that contains  without any explanation of this unusual and undefined terminology and notation cannot be qualified of "more readable of anything".
Therefore, I'll revert Wolfk.wk, and notify WT:WPM for some arbitration. D.Lazard (talk) 13:32, 12 September 2017 (UTC)
thread_title:  Section congruence and residue classes 

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thread_title:  External links modified 

message: Quote: "an ideal is a subset of a ring that is stable under addition and multiplication by any element of the ring". Under addition? No. Check the article about ideals: 'A subset  I is called a two-sided ideal (or simply an ideal) of  R if it is an additive subgroup of R that "absorbs multiplication by elements of R."'. A trivial counterexample is 2\mathbb{Z} in \mathbb{Z}, which is not stable under adding 1. 37.117.118.138 (talk) 10:59, 22 March 2018 (UTC)
replies: text: OK, the sentence was ambiguous, as "by any element of the ring" should apply only to multiplication. For avoiding the ambiguity, it should be written "an ideal is a subset of a ring that is stable under multiplication by any element of the ring and addition". But this is awkward, and "stable" is unnecessary jargon. Thus I have rewritten things for clarification. D.Lazard (talk) 11:24, 22 March 2018 (UTC)
thread_title:  Wrong