Talk:Gaussian integer

Definition

May I query this: "A Gaussian integer is a complex number whose real and imaginary part are both integers." That can't be right. The imaginary part is by definition some multiple of i. Even where b = 0, the imaginary part is a natural number, but still not an integer. Isn't it more correct to say "a complex number where the real part and the argument of the imaginary part are both integers"? JackofOz 01:53, 13 March 2006 (UTC)

Be careful. The imaginary part of 2+3i is 3, not 3i. Dmharvey 01:55, 13 March 2006 (UTC)

Also, but 0 is an integer, you may be confusing the integers with the positive integers. And you may be confusing modulus with argument, as the argument of any non-zero integer multiple of i is π/2, while its modulus is that integer which you multiplied by i. --PhiJ 16:37, 5 November 2006 (UTC)

Hi, I am confused by the Latex definition given on the page "Formally, Gaussian integers are the set "

$\mathbb {Z} [i]=\{a+bi\mid a,b\in \mathbb {Z} \}.$

Shouldn't $a,b\in \mathbb {I}$ ? I thought every complex number can be represented as a tuple of two real numbers. And these Gaussian integers ought to belong to $\mathbb {R}$ or to be precise the set $\mathbb {I}$. Also, the above query/talk/point/discussion mentions that both a, b are integers, while the Latex definition, as per my understanding, puts them in $\mathbb {Z}$. My apologies if I am wrong here as I dont understand this math concept. However, its precisely the reason for my doubt. I hope somebody with proper understanding clarifies this. -- wadkar <AT> gmail <DOT> com — Preceding unsigned comment added by 125.63.107.5 (talk) 22:03, 17 April 2012 (UTC)

I don't know what you mean by $\mathbb {I}$, but $\mathbb {Z}$ is standard notation for the set of (real) integers. It is short for the German word "Zahlen". —David Eppstein (talk) 22:46, 17 April 2012 (UTC)

Thanks, I got confused by the notation and their meaning. — Preceding unsigned comment added by 125.63.107.34 (talk) 17:07, 27 April 2012 (UTC)

Other kinds of "complex integers"

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)

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)

Relation to quadratic integers

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)

Repartition???

The first sentence in the section Unsolved problems is as follows:

"Most of the unsolved problems are related to the repartition in the plane of the Gaussian primes."

However, there is no explanation of what the word "repartition" means. Also, I've been around a while and have never seen the word "repartition" used in mathematics or anywhere else. Looking up the word in dictionaries did not help explain its use in this article.

I strongly suggest that this sentence be rephrased to avoid the use of that word. Or if for some reason it is important to use that word, then will someone please define it in that section, since I expect that at least 99.99% of readers of this article have no idea what it means.

Its meaning cannot be inferred from context, either. (Though this would be a very bad way for an encyclopedia article to communicate the meaning of a word.)

Note: This word is also used in the caption to the illustration of "Repartition in the plane of the small Gaussian primes" — which is just a picture of the small Gaussian primes — also giving no hint of what the word means. This is another place the word should not be used!!!Daqu (talk) 12:21, 22 April 2016 (UTC)

repartition n 1: DISTRIBUTION 2: a second or additional partition (Webster's Seventh New Collegiate Dictionary 1969)

The word has been in use for quite some time. — Anita5192 (talk) 18:46, 22 April 2016 (UTC)

If you think most users would not understand the word in this context, you could change it to something else. — Anita5192 (talk) 18:50, 22 April 2016 (UTC)

This is some translation awkwardness by a non-native speaker. Répartition in French mean distribution in English. So both uses are about the distribution of Gaussian primes, which makes a lot more sense. I'll make the change. --Mark viking (talk) 20:19, 22 April 2016 (UTC)

Gaussian Primes

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 Y_i such that X = ΠY_i? 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)

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)

Disputed question

Quotation: For all Gaussian primes except $p_{1}:=1+i$, there is one and only one associate, which fulfils the congruence $p\equiv 1{\pmod {2+2i}}$

disputed inline|reason= wrong: a+bi and -a–bi are congruent mod 2+2i and associated|date=August 2017

Your statement is wrong. What makes you think, that for an arbitrary Gaussian integer z=a+ib holds $z\equiv -z{\pmod {2+2i}}$? The definition (see chapter Congruences and residue classes below) says, that there must exist a factor $q$, with $z-(-z)=2z\;{\stackrel {!}{=}}\;q(2+2i)=2q(1+i)$, i.e. $z\;{\stackrel {!}{=}}\;q(1+i)$ This is only the case, if $(1+i)|z$. That again only holds for even Gaussian integers (defined in same chapter, see example 1), which are never primes, except the noted $p_{1}=1+i$ Please remove the dispute kind regards--Wolfk.wk (talk) 17:19, 13 August 2017 (UTC)

To editor Wolfk.wk: OK, my argument is wrong. But this is a very strange way for saying that every prime different of $1+i$ has exactly one associate $a+ib$ with a odd and positive and b even. As your way for choosing the prime element seems WP:OR, I'll change my tag into {{citation needed}}. D.Lazard (talk) 20:09, 13 August 2017 (UTC)

To D.Lazard

this is by no means a strange way, it is original research.

Please note: this was not my idea (I wish it was), but the idea of Mr. Gauß himself. Please look at page 546 of his paper (unfortunately in German), there you will find exactly my statement, given by Gauß:

H. Maser (Hrsg.): Carl Friedrich Gauss’ Arithmetische Untersuchungen über höhere Arithmetik. Springer, Berlin 1889, S. 534 ff.

I think I will include the citation in my next edit, and hope you believe it now...

Wolfk.wk (talk) 20:36, 13 August 2017 (UTC)

To D.Lazard

I have included the citation, please remove your tag citation needed.

--Wolfk.wk (talk) 09:57, 14 August 2017 (UTC)

The citation that has been provided shows that the choice of prime associate is the one that was given by Gauss. However, mathematics have evolved since Gauss, and a citation is still lacking for showing that this choice is standard in modern literature. In fact this choice is somehow problematic, as, for example, 231 has not the same factorization over the integers and the Gaussian integers (3.7.11 vs. (–1).(–3).(-7).(–11)). This could be acceptable if there were not a better (an easier) choice for the associate. In fact, every Gaussian integer with an odd norm has a unique associate a+ib, with a odd and positive (and thus b even). This way of choosing associates is easy to prove and gives the natural associate for real primes. It is thus certainly preferred by modern textbooks, even I have no source at hand for justifying that. Nevertheless, I agree that Gauss's choice should be mentioned, but only has an historical remark.

By the way, the previous version of the article was not only incomplete, but also badly structured. The recent edits do not improve the structure, and may be confusing, as the edits emphasize on the Gauss' original point of view, without any connexion with modern knowledge on the subject. The choice of the prime associates is an example. The description of the residue classes without any reference to the modern theory of lattices is another example. In a near future, I'll try to remediate these issues. D.Lazard (talk) 14:05, 20 August 2017 (UTC)

I agree, that there are a plenty of other possibilities to define 'primary', but I disagree, that others are better than Gauß' proposal, even if some might find it 'strange'. I will try to explain the problem.

The definition of primary associates which you propose, is surely possible. Yet another simple way is, to choose them from the first quadrant (excluding the imiginary axis), i.e.$a+bi$ with $a>0,b\geq 0$.

I am absolutely sure, that Gauß has considered all these ideas (he was a genius, as you probably know). His final definition is to prefer, because it fulfills an important requirement: the product of two primary numbers should also be a primary number. This is the case for his proposal, not for yours (and also not for the 'I. quadrant choice'). A simple counterexample: $(1+2i)^{2}=-3+4i$.

In fact, he gives yet another possibility in his his paper, which fulfills this requirement: $a+bi$ with $a\equiv 1{\pmod {4}}$ and $b$ even.

I have severe doubts, that a modern textbook could give a better explanation than Gauß himself, but you may convince me with a reference. I think, these topics are of such fundamental nature, that 'modern research' can hardly give any new insights.

Your assertion that the article is badly structured, is wrong. The article is clearly structured and contains most of the important & relevant topics about Gaussian integers.

By the way, I am very curious to hear, what lattices have to do with residue classes? If you add an extension to the article, I will surely read it with interest. --Wolfk.wk (talk) 07:34, 21 August 2017 (UTC)

Latest edits of D. Lazard

D. Lazard has deleted and rewritten major parts of the article and worsened it with this edits. He has deleted decent computations and examples and replaced them by wrong ones. The text is now much less readable and users will have difficulties to understand the ideas. I had a long going dispute with him on this page (see above), in which he was never able to prove his claims, but despite this he went on with his destruction. Here are two of the worst examples:

• Section Greatest common divisor:

I had written a explanation how to compute the gcd using Euclid algorithm step by step, which was easy to understand for readers and have given an instructive example for that. He deleted this completely and replaced it with useless methods to compute the gcd by using the norm. Here it is:

Deleted version:

Otherwise the Euclidean algorithm can be used: For the determination of the gcd of two Gaussian integers $z_{0},z_{1}$ it works very similar as for real integers.

It holds $(z_{0},0)=z_{0}$ for all $z_{0}$ (especially $(0,0)=0$). And for $z_{1}\neq 0$ there exists a pair of Gaussian integers $q_{1},z_{2}$ with

$z_{0}=q_{1}z_{1}+z_{2}$ and $|z_{2}|<|z_{1}|.$

To get this, one takes $q_{1}$ as a Gaussian integer lying next to the quotient $\xi :={\frac {z_{0}}{z_{1}}}$ (there may be up to four). Then always holds $\left|q_{1}-\xi \right|\leq {\frac {1}{\sqrt {2}}}$ (see above) and consequently $|z_{2}|\leq {\frac {|z_{1}|}{\sqrt {2}}}$.

If $z_{2}\neq 0$, this is continued with $z_{1}=q_{2}z_{2}+z_{3}$ and $|z_{3}|<|z_{2}|$ a.s.o, until finally $z_{n+1}=0$. It is easy to see, that then $z_{n}$ is the sought-after gcd: $(z_{0},z_{1})=z_{n}$. Example: Sought-after shall be the gcd of $z_{0}=5+\mathrm {i} ,\;z_{1}=2$. The quotient is ${\frac {z_{0}}{z_{1}}}=2.5+0.5\mathrm {i}$. For $q_{1}$ therefor the four Gaussian integers $2,2+\mathrm {i} ,3,3+\mathrm {i}$ can be chosen. We chose e.g. $q_{1}=2$ and get $z_{2}=z_{0}-q_{1}z_{1}=5+\mathrm {i} -4=1+\mathrm {i}$. The next step gives ${\frac {z_{1}}{z_{2}}}={\frac {2}{1+\mathrm {i} }}=1-\mathrm {i}$, i.e., the residue is $z_{3}=0$: The algorithm terminates and we get as gcd ${\underline {(5+\mathrm {i} ,2)=z_{2}=1+\mathrm {i} }}$.

Lazard's replacement:

This algorithm consists of replacing of the input (a, b) by (b, r), where r is the remainder of the Euclidean division of a by b, and repeating this operation until getting a zero remainder, that is a pair (d, 0). This process terminates, because, at each step, the norm of the second Gaussian integer decreases. The resulting d is a greatest common divisor, because (at each step) b and r = a – bq have the same divisors as a and b, and thus the same greatest common divisor.....It consists in remarking that the norm N(d) of the greatest common divisor of a and b is a common divisor of N(a), N(b), and N(a + b). When the greatest common divisor D of these three integers has few factors, then it is easy to test, for common divisor, all Gaussian integers with a norm dividing D.

Ingoring the error, that there should stand N(ab) instead of N(a + b), this is unusable for more complicated cases. Simple counterexample: the gcd of 3+2i and 2+3i is not 3+2i (nor 2+3i) what his formula gives (since both have N(3+2i) = N(2+3i) = 13) but 1!!

• Section Congruences and residue classes

He claims: As the area of this square is N(m), it contains exactly m Gaussian integers.... Ignoring the mistake, that he should have written N(m) instead of m, this is definitely not true. Not every square of area N contains N grid points, it may be more or less. A counterexample, which is easy to see: A square of the area 3 may contain 1, 2, 3 or 4 gridpoints, depending on its position. I had given a simple geometrical proof for the correct statement, which he has simply deleted. The explanation of the figure, which I provided Wikipedia, is also simply removed. No one can understand now, what it means.

There are many other mistakes, and I suggest that these latest edits shoud be undone. Could somebody of the page watchers please help?

--Wolfk.wk (talk) 07:30, 4 September 2017 (UTC)

"Ingoring the error, that there should stand N(ab) instead of N(a + b) ...: This is not an error, as the gcd of a and b divides also a + b (this is the basis of the proof of Euclidean algorithm). Not ignoring the error, we have N(3+2i) = N(2+3i) = 13 and N(a +b) = 50, which have the gcd 1, showing that the gcd of 3+2i and 2+3i divides 1 and is therefore one.

"As the area of this square is N(m), it contains exactly m Gaussian integers.... Ignoring the mistake, that he should have written N(m) instead of m, this is definitely not true": I have clarified in the article that it is a semi-open square that is considered here. For this square, the result is true. I agree that either a proof or a link to the article about lattice would be useful, and I'll add this in a next edit. As the proof is standard in lattice theory, giving detailed proof mixed in a difficult-to-understand geometric description, and not related with lattice theory is misleading and must be avoided. D.Lazard (talk) 13:00, 4 September 2017 (UTC)

I have not found in Wikipedia articles the basic result of lattice theory which implies that the number of residue classes is the norm of the modulus. Therefore I have provided one essentially derived from Wolfk.wk's one, but, as this is not really specific to Gaussian integers, I have put it in a collapsed box. D.Lazard (talk) 21:34, 4 September 2017 (UTC)

General omments: Before Wolfk.wk's edits, the article was incomplete and badly structured. In particular, although almost every properties result from Euclidean division, this was sketchy described after the description the resulting properties. Also, these properties were systematically presented in terms of abstract ring theory, which is unnecessarily WP:TECHNICAL. Also, the fundamental result, that the norm is the number of residue classes, was completely lacking. Wolfk.wk's edits have the merit of filling some gaps in the previous version. However, they suffers of several issues. Firstly they limit them to Gauss' original terminology without any link to modern terminology. Also, although Gaussian integers are the basic example for learning algebraic number theory, the distinction was unclear between the properties that was specific to Gaussian integers and the more general properties.

So Wolfk.wk's edits were an improvement, but were not fully satisfactorily. These are the reasons for which I have rewritten the article. By doing this I have removed some details, because they do not seem really useful. For example, detailed examples for the application of Euclidean algorithm seem not useful, as this duplicates (except for the sub algorithm of Euclidean division) the article Euclidean algorithm. On the other hand the use of the norm for improving gcd computation is specific to Gaussian integers (and other rings of algebraic integers), and this deserve to be exemplified (I have given such an example, but others could be useful). Also, examples for Euclidean division could be useful, and Wolfk.wk's example could be useful for that, if adapted for this purpose. This example is not convenient for gcd, as the use of the norm makes it completely trivial. There are certainly more possible improvements of my version, but, as it is, I am convinced that it is better than all preceding versions. D.Lazard (talk) 21:34, 4 September 2017 (UTC)

My comments:

1. GCD: OK, if one uses 'N(a+b)' as third term, my counterexample above does not work. Anyway, your method is not really satisfactory, since it gives only a guess (all Gaussian integers of given norm), an the user has to test all of them anyway, by dividing the given numbers explicitely. Your description of of Euler's algorithm is in my opinion also worse than the previous step-by-step explanation, and I will restore this and the example in the next time.
2. Number of gridpoints: After long dispute you have finally realised, that your claim was wrong, and my proof (which you had deleted) is needed. You have rewritten it, but IMO less explanative and in poorer English. Therefore, I will probably restore the previous version in the next time.

Your action in the latest edits was not acceptable. You don't have the right to delete decent content and replace it with one, that is wrong or less readable, with the specious justification to make it more encyclopedic.

Please stop acting, as if you were the owner of this page. --Wolfk.wk (talk) 08:55, 5 September 2017 (UTC)

Please read WP:Civility, and think about it. "Your action in the latest edits was not acceptable. You don't have the right to delete ...": On which Wikipedia rules are based these assertions? Moreover the fact that your added content was decent and acceptable is just your personal opinion, nothing more. D.Lazard (talk) 09:36, 5 September 2017 (UTC)

These assertions are based on the rules of fairness and respect for the work of co-workers, which are surely also anchored somewhere in WP. Your last sentence is right, but you should acknowledge that it applies for you, too. --Wolfk.wk (talk) 08:08, 6 September 2017 (UTC)

Especially it also applies to your General comments above: Your statements are just your personal opinion, nothing more. --Wolfk.wk (talk) 18:22, 8 September 2017 (UTC)

Section congruence and residue classes

Wolfk.wk 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).

Wolfk.wk has just restored his version with the edit summary: Restored the version prior to Lazard's deteriorations, which destroyed the context for the figure. This version is also much more instructive and better readable (discussion see 'talk page'). 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 "The grid shall consist of the lines with $s=\pm {\frac {1}{2}},\pm {\frac {3}{2}},\dots ,\;t=-\infty ,\dots \infty$ and $t=\pm {\frac {1}{2}},\pm {\frac {3}{2}},\dots ,\;s=-\infty ,\dots \infty$" 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)

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Wrong

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)

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)