Questions tagged [gaussian-integers]

This tag is for questions relating to the Gaussian integer, which is a complex number $~z=~a~+i~b~$ whose real part $~a~$ and imaginary part $~b~$ are both integers.

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How do I use the gaussian divisors formula?

For an integer z, $$ z = \epsilon \prod_i p_i^{k_i}, $$ where $\epsilon$ is and a unit and every $p_i$ is a Gaussian prime in the first quadrant then the sum of the Gaussian divisors is $$ \sigma_1 (z)...
codecademic800's user avatar
-3 votes
2 answers
52 views

Real numbers extend to complex numbers. Why do (real) integers extend to 'Gaussian integers' instead of 'complex integers'? [closed]

Real numbers extend to complex numbers are $\mathbb C := \{ a+bi;a,b \in \mathbb R \}$. So why isn't the set $\{a+bi;a,b \in \mathbb Z \}$ called complex integers instead of Gaussian integers, an ...
BCLC's user avatar
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prime factorization in $\mathbb{Z}[i]$ [duplicate]

We were asked to show where the following reasoning goes wrong. Since $1+i$ and $1-i$ are prime elements in $\mathbb{Z}[i]$, the equation $$(-i)(1+i)^2=(1+i)(1-i)=2$$ show that unique prime ...
riescharlison's user avatar
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31 views

Cauchy principal value for a Gaussian integral of a rational function

I wish to calculate integrals of the following form, all Gaussian integrals of a rational function, on the entire real domain: $$ I_{k,n} = \int_{-\infty}^{\infty} dx \frac{e^{-x^2/2}}{\sqrt{2\pi}} \...
Uri Cohen's user avatar
  • 375
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0 answers
6 views

Zariski-density on almost diagonal embedding

It is not hard to see that the Gaussian integers $\mathbb{Z}[i]$ are Zariski-dense inside $\mathbb{C}$, seen as an affine space over $\mathbb{C}$. Consider now the set $$D = \{(z,\overline{z}) \in \...
Henrique Augusto Souza's user avatar
9 votes
2 answers
164 views

Remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\Bbb{Z}[i]$

Find the possible remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\mathbb{Z}[i]$. I'm having a hard time understanding remainders in $\mathbb{Z}[i]$. I'm gonna write my solution to the ...
huh's user avatar
  • 387
2 votes
1 answer
62 views

Question about the prime elements of $\mathbb{Z}[i]$.

I am learning about gaussian integers and I have a few questions about the following argumentation. What are the prime elements in $\mathbb{Z}[i]$? We remember that only the units $+1,-1,+i,-i$ in $\...
NTc5's user avatar
  • 37
0 votes
1 answer
38 views

Ring structure of $\mathbb{Z}/(n^p+1)\mathbb{Z}$

I encountered the ring $\mathbb{Z}/(n^p+1)\mathbb{Z}$, where $n$ is some positive integer and $p\in\{2,3,5,...\}$ a prime and I am wondering whether there is a difference in the structure when $p=2$ ...
Jfischer's user avatar
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Prime Divisor of the Sum of Two Squares

I'm struggling something immensely to make sense of the following: https://meiji163.github.io/post/sum-of-squares/#sums-of-two-squares Factoring an integer in Gaussian integers is closely related to ...
StormyTeacup's user avatar
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Need help in understanding a proof of the fact: The set of Gaussian integers is a euclidean domain.

Let $J[i]$ denote the set of Gaussian integers. Show that $J[i]$ is a Euclidean ring. The proof given is as follows: In order to show this we must first introduce a function $d (x)$ defined for every ...
Thomas Finley's user avatar
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0 answers
75 views

Gaussian Integers how to solve with association

I have this problem following. Two elements $a$ and $b$ of a commutative ring $\mathbb{R}$ with one are said to be associated if there is a unit $u$ in $\mathbb{R}$ such that $a = u \cdot b$ Find $a \...
john's user avatar
  • 195
1 vote
0 answers
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How can i determine u here [closed]

The Gaussian integer $3-5i$ can be written as a product $3-5i = u\times p\times q$ , there u is a unit and and p,q are Gaussian primes in standard form. Decide u. How can i determine u here by ...
mire12's user avatar
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1 answer
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Question about $4n^2+4n+1$ in showing that $\mathbb{Z}[i]/I$ is finite?

Background Definition: The function $N:\mathbb{Z}[\sqrt d] \to \mathbb{Z}$ given by $$N(a+b\sqrt d)=(a+b\sqrt d)(a-b\sqrt d)=a^2 - db^2$$ is called the norm. Exercise 39: Let $I$ be a nonzero ideal ...
Seth's user avatar
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Questions about two proofs for showing $\mathbb{Z}[i]/I$ is finite, without using concept of norm for gaussian integers

Background: Theorem 6.6 Let $I$ be an ideal in a ring $R$ and let $a,c\in R,$ Then $a\equiv c \pmod I$ if and only if $a+I=c+I.$ Theorem 10.8 Every Euclidean domain is a principal ideal domain. ...
Seth's user avatar
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2 answers
98 views

Show that the GCD of 3+4i and 7-i in Z[i] is 1 [duplicate]

I have tried proceeding by starting dividing 3+4i by 7-i and the other way as well, but don't know how to get to 1
vaib's user avatar
  • 1
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1 answer
173 views

Show that: exists unique ring homomorphism $\varphi : \mathbb{Z} [ i ] \longrightarrow R$ with $\varphi ( i ) = a$, where $a^{2} =-1_{R}$

Problem: Let R be a ring which has an element $a \in R$ such that $a^{2}=-1_{R}$. Prove that: there exists a unique ring homomorphism $\varphi : \mathbb{Z} [ i ] \longrightarrow R$ such that $\varphi (...
TrItOs's user avatar
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What is the factorization of 26 in $\mathbb Z[i]$ into irreducible?

Here is the question I am trying to solve: What is the factorization of 26 in $\mathbb Z[i]$ into irreducible? My idea is: 26 = (5-i)(5+i) but it seems like this is an incomplete answer. Could anyone ...
Emptymind's user avatar
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60 views

Automorphism groups of $\mathbf Z$, $\mathbf Z[i]$, $\mathbf Z[\omega]$

I would like to know the automorphism groups of the rational integers $\mathbf Z$, the Gaussian integers $\mathbf Z[i]$, and the Eisenstein integers $\mathbf Z[\omega]$. My question is, would $\text{...
node196884's user avatar
1 vote
0 answers
85 views

Euler Phi Function for Gaussian integers

I am considering the Euler totient function for Gaussian integers. In reference to this question, I would wish to use the fact that $\phi(p^{k})=N(p)^{k−1}\phi(p)$ if $p$ is prime, but have not ...
V. Elizabeth's user avatar
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90 views

Gaussian integers relatively prime to $9$

I wish to count all Gaussian integers (modulo $9$) that are relatively prime to $9$. So far, I can easily do this for any Gaussian prime, by drawing the fundamental parallelogram and identifying all ...
V. Elizabeth's user avatar
1 vote
0 answers
73 views

Complete Residue System Modulo $z$ in the Gaussian Integers

Let $z \neq 0$ be a non-unit Gaussian integer. I wish to prove that the Gaussian integers in the fundamental parallelogram associated to $z$ are a complete residue system modulo $z$. I have succeeded ...
V. Elizabeth's user avatar
0 votes
2 answers
130 views

Proving $(x - i)$ and $(x + i)$ are coprime in $\mathbb{Z}[i]$

Edit: I misread the text, which says before the following passage that we assume $x$ to be even. In which case the statement is true (I believe). I am trying to understand a proof that for any [edit: ...
Anakhand's user avatar
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1 answer
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Diophantine Equation in Gaussian Integers [closed]

I am considering the Diophantine Equation $X^2 + 4Y^2 = 3^n$, for some fixed $n$. I would wish to find Gaussian solutions to this Diophantine equation, by using a prime factorisation argument in the ...
V. Elizabeth's user avatar
6 votes
1 answer
142 views

How many units are there in $\mathbb{Z}[i]/ (13^{2021})$?

Clearly, there are $(13^{2021})^2$ elements in $\mathbb{Z}[i]/(13^{2021})$ and I wish to determine how many of these are units. There are $\phi(13^{2021})=13^{2021}-13^{2020}=13^{2020}\cdot 12$ ...
Kadmos's user avatar
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1 vote
1 answer
121 views

Proving $|ad-bc|=1$ for the Product of Two Numbers Expressible as Sums of Two Squares

If $p$ is an odd prime with $p \equiv 1(4)$, then it can be uniquely expressed by the sum of two squares, i.e., $p=a^2+b^2$ with $gcd(a,b)=1$. We consider the integer u,v with form $1,2,p^k,2p^k$. The ...
kyle1117's user avatar
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0 answers
61 views

Using Chinese remainder theorem for Gaussian integers

I have the following two questions that am trying to solve. Use Chinese remainder theorem to solve this system of congruence $x \equiv i \mod (1+i)$ and $x \equiv 2 \mod 3i$. $x \equiv (−3+i) \mod ...
affkoff's user avatar
0 votes
0 answers
49 views

Finding every solution of $a^2+b^2+c^2=3$ in $\mathbb{Q}(i)$

Specifically, $a,b,c\in\mathbb{Q}(i)$ are complex numbers with rational parts whose squares sum to $a^2+b^2+c^2=3$. There's an answer to this question over $\mathbb{Q}$ already here but I couldn't ...
Chris Wolird's user avatar
1 vote
0 answers
152 views

Explicitly finding the elements of $\mathbb Z[i]/\langle 3+2i\rangle$ [duplicate]

Find the elements of $\mathbb Z[i]/\langle 3+2i\rangle$ It is a theorem that $\mathbb Z[i]/(a+bi)$ is isomorphic to $\mathbb Z/(a^2 + b^2)$ if $a,b$ are coprime, so we know $\mathbb Z[i]/\langle 3+2i\...
shintuku's user avatar
2 votes
1 answer
57 views

Residues and perfect squares in $\mathbb{Z}[i][X]$

Let $P \in \mathbb{Z}[i][X]$ be such that $P(0) = 1$ and $P\mod4$ is a perfect square in $(\mathbb{Z}[i] / 4\mathbb{Z}[i])[X]$. Moreover if needed we may assume $P$ is monic. Is there some way to ...
porridgemathematics's user avatar
0 votes
0 answers
56 views

Calculating modular roots over Gaussian integers

Let $a+bi$ be a Gaussian integer. Given another Gaussian integer $c+di$ how does one find $x^2\equiv(c+di)\bmod(a+bi)$? Can you illustrate with $x^2\equiv 48\bmod(156\pm89i)$?
Turbo's user avatar
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0 answers
68 views

question about gcd in Z[i] gauss integers [duplicate]

I'm studying $\mathbb{Z}[i]$, Gauss' integers, which I know is an euclidean ring. I want to prove that if $x,y\in\mathbb{Z}$ are such that $x\wedge y=1$, one being odd and the other even, and $\omega =...
Rafaël's user avatar
  • 181
4 votes
0 answers
62 views

Generalizations of the Gauss circle problem

The Wikipedia article on the Gauss circle problem states that, if there are $N(r)$ lattice points of radius less than $r$ from the origin, then we know that $|E(r)|=|N(r)-\pi r^2|=O(r^{131/208})$. Do ...
George Bentley's user avatar
4 votes
1 answer
134 views

Is $SL_2(\mathbb{Z}[i])$ finitely generated?

This may be a naïve question, but I have been thinking about the structure of the the matrix group $SL_2(\mathbb{Z}[i]) \subset SL_2(\mathbb{C})$. One thing that has been on my mind is whether or not $...
An Isomorphic Teen's user avatar
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0 answers
118 views

If $p\in\mathbb{Z}$ is a prime and $p\equiv 1$ mod $4$, show that the quotient ring $\mathbb{Z}[i]/(p)$ has order $p^2$.

The Problem: Let $p\in\mathbb{Z}$ be a prime with $p\equiv 1$ mod $4$. Show that $\mathbb{Z}[i]/(p)$ has order $p^2$. Source: Abstract Algebra $\mathit{3^{rd}}$ edition by Dummit and Foote. My Attempt:...
Dick Grayson's user avatar
  • 1,393
1 vote
1 answer
253 views

$\Bbb Z[i]$ is a principal ideal domain (Marcus' Number Fields, Exercise $1.7$)

I want to show that $\Bbb Z[i]$ is a principal ideal domain, following the hints in Marcus's Number Fields. Let $I$ be an ideal in $\Bbb Z[i]$, and choose $\alpha \in I\setminus\{0\}$ such that $N(\...
stoic-santiago's user avatar
1 vote
1 answer
63 views

Investigating number of equivalence classes in the Gaussian integers formed by adding integers.

I'm trying to find how many equivalence classes on the Gaussian integers can be formed just by adding integers (as part of a wider consideration on how many there are altogether). Let $\gamma \in \...
Ncrest's user avatar
  • 23
3 votes
0 answers
60 views

Quadratic Sieve on the Gaussian Integers

Factoring a large Gaussian integer $z_0 = a+bi$ into Gaussian primes may be done by first factoring the norm $N(z_0) = a^2 + b^2$ over the integers, and then considering the factors of each integer ...
Samuel Li's user avatar
  • 785
5 votes
1 answer
145 views

Why do the Gaussian rationals make these patterns?

I was looking for a cool thing to visualize, when I found this picture on Wolfram MathWorld. I obtained the image below by taking all integers $a, b, c, d$ in a given range (between $-15$ and $15$ ...
zenzicubic's user avatar
0 votes
0 answers
41 views

Let $p\in\mathbb{N}$ be prime. Prove that p is irreducible in $\mathbb{Z}[i]$ if and only if $p\equiv3$ $(mod$ $4)$. [duplicate]

I have already proven one direction, which is $p\equiv3$ $(mod$ $4)$ implies $p$ irreducible. Now, stating that $p$ is irreducible I can't get to the conclusion that $p\equiv3$ $(mod$ $4)$. I have ...
Pau Vallès Martínez's user avatar
2 votes
2 answers
120 views

Number of ring homomorphisms

I'm trying to solve the following problem: Determine how many different ring homomorphism from $\mathbb{Z}[i] \to \mathbb{Z}/(85)$ exist. For a previous question I had to determine the unique integer ...
nymphodor's user avatar
4 votes
2 answers
289 views

What is the motivation behind the concept of definition of Gaussian prime numbers?

What is the motivation behind the concept of definition of Gaussian prime numbers? I am interested about the part of the definition when both real and imaginary coefficients are non-zero. Then a ...
Janik's user avatar
  • 89
0 votes
0 answers
69 views

Prove that the incomplete quotient and the remainder at division in $\mathbb {Z} [i]$ are defined ambiguously.

I was told that it is possible to understand this by finding at least two ways to divide with a remainder $3+2i$ by $1+i$ (ex. $(3+2i) \div (1+i)$), but i don't really understand how this will help me....
Andrew's user avatar
  • 1
2 votes
1 answer
177 views

Is there an effective way to decompose gaussian integers into prime factors?

We define $\mathbb{Z}[i] := \{a + bi \mid a, b \in \mathbb{Z}\}, i = \sqrt{-1},$ which is an euclidean ring together with $N: \mathbb{Z}[i] \to \mathbb{N}_0, z \mapsto z\bar{z}=a^2+b^2$ for $z=a+bi$. ...
jupiter_jazz's user avatar
2 votes
1 answer
115 views

NOT a maximal ideal of the ring of Gaussian integers

I have to show that $I= \{ a+bi : a,b \in \mathbb{Z} , a=5m , b=5n\}$ is NOT a maximal ideal of $\mathbb Z[i]$. For this if I take $J= \{ a+bi : 5\ $divides $ a^2+b^2 \}$ then , I can have that $4+...
Sharmi C's user avatar
  • 419
1 vote
1 answer
60 views

Determine the ideal $\langle 2i \rangle$ of the Gaussian integers $\Bbb Z[i]$. Describe the quotient $\Bbb Z[i]/\langle 2i\rangle$.

Determine the ideal $\langle 2i \rangle$ of the Gaussian integers $\Bbb Z[i]$. Describe the quotient $\Bbb Z[i]/\langle 2i\rangle$. I am having confusion with the theorem for $\langle 2i \rangle$. It'...
Podelora's user avatar
2 votes
1 answer
46 views

Finding three perfect complex powers with equal norm and cancelling imaginary parts

The Problem I've been curious about finding 3 distinct Gaussian integers with equal norm whose imaginary parts cancel out. So $\alpha, \beta, \gamma\in\mathbb{Z}[i]$ with $|\alpha|=|\beta|=|\gamma|$ ...
WhiteStoneJazz's user avatar
2 votes
1 answer
124 views

What is the prime factorization $6+12i \in \Bbb Z[i]$?

What is the prime factorization $6+12i \in \Bbb Z[i]$? I'm reading an abstract algebra sheet where they had the following problem, but I cannot find any examples on irreducible elements of $\Bbb Z[i]$...
Walker's user avatar
  • 1,404
1 vote
0 answers
81 views

Expressing integers as the sum of the squares of two Gaussian integers

I proved that it is possible to express any integer as the sum of the squares of two Gaussian integers (see proof below). My question is whether or not there are other simpler proofs. My proof: For ...
Will Octagon Gibson's user avatar
1 vote
1 answer
94 views

Equating coefficients of $ \sum_{n \ge 1} \frac{a_n}{n^s} = \sum_{n \ge 1} \frac{b_n}{n^s}, $ [duplicate]

Given that $$ \sum_{n \ge 1} \frac{a_n}{n^s} = \sum_{n \ge 1} \frac{b_n}{n^s}, $$ for all complex values of $s$ and where $a_n$ and $b_n$ are non-negative integers, can we say that $a_n = b_n$ for ...
Arpita Korwar's user avatar
3 votes
3 answers
118 views

Absolute convergence of $\sum_{n,m\geq 1}\frac{1}{(n+im)^3}$ related to the sum of squares function

As stated in the title, the series $\sum_{n,m\geq 1}\frac{1}{(n+im)^3}$ is absolutely convergent, if and only if $$ \sum_{n,m\geq 1}\frac{1}{(n^2+m^2)^{3/2}}=\sum_{N\geq 2}\frac{r_2(N)}{N^{3/2}}<\...
Christoph Mark's user avatar

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