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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
74 views

Prime ideals of subrings of the ring of Gaussian integers

Can anyone give me a hint with proving the following, Let $\alpha\in\mathbb Z[i],$ and let $P$ be a non-zero prime ideal of $\mathbb Z[\alpha].$ Show that the quotient $\mathbb Z[\alpha]/P$ is a ...
user avatar
0 votes
0 answers
32 views

Find any or all Lipschitz quaternions corresponding to a given quaternion norm

In this answer, in response to the question "How to find $2+7𝑖$ from $53$?", user @Cocopuffs provided a reference to a Jacobsthal sum, which enables obtaining a Gaussian integer, based on a ...
user avatar
  • 19
0 votes
0 answers
23 views

Cardinality of an euclidean ring [duplicate]

We want to show that the cardinality of the field $\frac{\Bbb Z[i]}{(3)}$ is equal to $9$. A similar exercise consists of showing that the cardinality of $\Bbb Z/3\Bbb Z $ is $3$, to that we can say ...
user avatar
  • 996
0 votes
0 answers
20 views

DFT, but with large values available

Suppose that we are calculating a size $N$ integer-valued DFT, with some values possibly adjoined to the integers, such as the imaginary $i$. My question is, if the word size allows integers much ...
user avatar
2 votes
1 answer
41 views

Difficulty with Complex Computation in Proof on Ford Spheres

I am reading a paper on arXiv taking the idea of Ford circles and putting it into three dimensions, but am having difficulty with the following lemma, at a very basic level. Most of the details aren't ...
user avatar
  • 1,569
0 votes
0 answers
32 views

Factor ring of Gaussian integers is a field

I want to show that $E=\mathbb{Z}[i]/\langle2-i\rangle$ is a field. To do this, I note that $R/I$ is a field iff $I$ is a maximal ideal. Moreover, a maximal ideal is a prime ideal in a commutative ...
user avatar
0 votes
0 answers
18 views

Factor ring is a field and finding its characteristic [duplicate]

Let $E = \mathbb{Z}[i]/I$ where $\mathbb{Z}[i]$ denotes the ring of Gaussian integers and $I=\langle 2-i \rangle$, an ideal of $\mathbb{Z}[i]$. We need to prove that $E$ is a field and then go on to ...
user avatar
1 vote
0 answers
37 views

Complex analogue of the golden ratio

Golden ratio $\phi$ and related numbers give the largest errors when approximated by rational numbers. I imagine that if we consider approximations of complex numbers by gaussian rationals or ...
user avatar
  • 321
3 votes
1 answer
91 views

Quotient Ring of Gaussian Integers - $\mathbb{Z}[i]/I_n$ is a field iff $q = n^2 + 1$ is prime.

I am working on a problem in which I need to prove that $\mathbb{Z}[i]/I_n$ is a field if, and only if, $q = n^2 + 1$ is prime. Here, $I_n = _{\mathbb{Z}[i]}\langle i -n \rangle$ is a given ideal of ...
user avatar
  • 150
1 vote
0 answers
41 views

Let $\mathbb{Z}[i]$ denote the *Gaussian integers*. Factor both $3+i$ and its norm into primes in $\mathbb{Z}[i]$

Question: Let $\mathbb{Z}[i]$ denote the Gaussian integers. (a) Compute the norm $N(3+i)$ of $3+i$ in $\mathbb{Z}[i]$ (b) Factor both $3+i$ and its norm into primes in $\mathbb{Z}[i]$ (c) Compute $\...
user avatar
  • 1,151
0 votes
1 answer
112 views

Surjective ring homomorphism from $\mathbb{Z}[i] \to \mathbb{Z}_q$

So I've got this question that is asking me to show that the map $\phi: \mathbb{Z}[i] \to \mathbb{Z}_q$ such that $\phi(r + is) = [r] +[s][n]$ is a surjective ring homomorphism where $0 \neq n \in \...
user avatar
0 votes
1 answer
37 views

Lemma about primeness of Gaussian integer

I don't understand the proof of this lemma: Let $p>1$ be an integer prime number. If $p$ is not prime gaussian integer then $p=\pi\bar{\pi}$ where $\pi$ and $\bar{\pi}$ are prime and conjugate in ...
user avatar
1 vote
0 answers
43 views

Gaussian primes with the same norm

I have the next question. Let be $F=\mathbb{Q}[i]$. Also let $\alpha=a+bi$ and $\beta=c+di$ be non associate primes in $\mathbb{Z}[i]$ such that $N(\alpha),N(\beta)\equiv\;1\;(\mathrm{mod}\;4)$ and $N(...
user avatar
  • 653
1 vote
1 answer
58 views

Gaussian primes $\alpha$ and $\beta$ with $\mathrm{gcd}(\mathrm{Im}(\alpha),\mathrm{Im}(\beta))=1$

Let be $F=\mathbb{Q}[i]$. Also let $\alpha=a+bi$ and $\beta=c+di$ be primes in $\mathbb{Z}[i]$ such that $N(\alpha), N(\beta)\equiv\;1\;(\mathrm{mod}\;4)$ and $N(\alpha)\not=N(\beta)$ I am trying to ...
user avatar
  • 653
2 votes
1 answer
73 views

Homomorphism ring from integers to quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. Let $\phi: \mathbb{Z} \rightarrow \mathbb{Z}[i]/(2+3i) \text{ where } \phi(z) = z + (2+3i)\mathbb{Z}[i]$. Because I want to prove that $...
user avatar
0 votes
1 answer
30 views

Uniqueness of expansions of Gaussian integers to a Gaussian integer base

Let $\beta$ be a Gaussian integer whose norm $N$ is at least three, and let $u_1,\ldots,u_N$ be distinct Gaussian integers at least one of which is not divisible by $\beta$. Consider the set $R$ of ...
user avatar
0 votes
0 answers
116 views

The Diophantine equation $x^2\pm iy^2=z^3$ in Gaussian integers

I am trying to find any reference about the solutions of the Diophantine equation $x^2\pm iy^2=z^3$. It seems to me that I read about this somewhere before, but I can't remember where. Any ...
user avatar
2 votes
1 answer
43 views

Gaussian numbers algorithmic operations.

I have been working with this idea For some time I am struggling for a particular case, but first I will give the idea of this. Let be $F=\mathbb{Q}[i]$ and let be $\alpha\in F$, therefore $\alpha=\...
user avatar
  • 653
0 votes
0 answers
41 views

If $p$ is a prime in $\mathbb{Z}$ and $(p)$ is a prime ideal in $\mathbb{Z}[i]$, then $x^2 \equiv -1\pmod{p}$ has no solution in $\mathbb{Z}[i]$

I can't find the proof for this. I know that $\mathbb{Z}[i]/(p)$ is an integral domain, so I tried to assume that there is a solution for $x \in \mathbb{Z}[i]$ and reach a contradiction of that fact, ...
user avatar
0 votes
0 answers
21 views

Generator/generators for cyclic group $\mathbb Z[i]/(11-8i)$, given the following isomorphism: [duplicate]

I need to find a generator for $V$, being $V$ the $\mathbb Z[i]$-module generated by $v_1,v_2$ where: $(1+i)v_1 + (2-i)v_2 = 0$ $3v_1 + 5iv_2 = 0$ [EDIT] To clarify $V = \frac{\mathbb Z[i]^2}{A\mathbb ...
user avatar
0 votes
0 answers
73 views

Proof that gaussian integers are an integral domain by contradiction.

I just want to make sure my proof is alright, I would appreciate some hints for a direct proof also. What I have doesn't feel very elegant. Assume for contradiction that $\alpha, \beta$ are zero-...
user avatar
0 votes
0 answers
40 views

Find the factorization of the following numbers in Z[i]. 11 + i ; 510 + 180i:

The factorization of the following numbers in Z[i]. 11 + i 510 + 180i Usually to determine the factorization of a number x we would find its prime factorization say $x=y*z$. Then, we would determine ...
user avatar
0 votes
2 answers
86 views

How do I find $a,b\in\mathbb{Z}$ s.t. $\{ac-bd+i(ad+bc)\mid c, d\in\mathbb{Z}\}$ have real and imaginary parts both even or both odd?

I'm trying to find some numbers, $a,b\in\mathbb{Z}$ s.t. the following equation is satisfied. \begin{equation} \{a c-b d+i(a d+b c) \mid c, d \in \mathbb{Z}\}=\{k+i l \mid k, l \text { even or odd }\} ...
user avatar
  • 409
1 vote
1 answer
41 views

How do I find $a,b \in \mathbb{Z}$ s.t. $\{a c+b d+i(a d+b c) \mid c, d \in \mathbb{Z}\}$ consist of cmplx no. with real,imag part even or odd?

I'm trying to find some numbers, $a,b\in\mathbb{Z}$ s.t. the following equation is satisfied. \begin{equation} \{a c+b d+i(a d+b c) \mid c, d \in \mathbb{Z}\}=\{k+i l \mid k, l \text { even or odd }\} ...
user avatar
  • 409
0 votes
0 answers
32 views

A question about Gaussian integers [duplicate]

I am working on the following exercise: Show that if $p \equiv 1 \mod 4$ then $p$ is not prime in $\mathbb{Z}[i]$, but instead splits as the product of two distinct prime. [Hint: Show that $p|(a^2+1)$ ...
user avatar
0 votes
1 answer
99 views

Prove that the Gaussian integers Z [i] form an integral domain

This is all I have done. Is this correct? please explain. Proof. 0 = 0 + 0i is in Z [i] : Also note that 1 = 1 + 0i is in z[i]. Then (a + bi)(c + di) = (ac) + (bd)i is in Z [i] and (a + bi)(c + di) = (...
user avatar
0 votes
0 answers
39 views

What are the prime ideals in $\mathbb{Z}[i]$? [duplicate]

Let $p \in \mathbb{Z}$ be prime. I am trying to determine all prime ideals $\mathfrak{q} \subset \mathbb{Z}[i]$ such that $\mathfrak{q} \cap \mathbb{Z} = p\mathbb{Z}$. I know that $$ \{\text{prime ...
user avatar
2 votes
1 answer
67 views

In calculating GCD in $\mathbb Z[i]$ does it matter if we switch $a$ and $b$?

I am trying to solve this problem: Find the generator of the ideal $(47 - 13i, 53 + 56i).$ I know that I should use Euclidean Algorithm but I am wondering if it matters if I divided $a = 47 - 13i$ by $...
user avatar
  • 1,187
0 votes
0 answers
48 views

Implications of Modular Arithmetic in the Complex Plane Defined as $z_1 \bmod z_2 = z_1 - z_2 \left\lfloor \frac{z_1}{z_2} \right\rfloor$?

Previously, I found $a \bmod b = a - b \left\lfloor \frac{a}{b} \right\rfloor$ and wondered if this could be extended to the complex plane. I did this, and it seems to yield interesting results, such ...
user avatar
0 votes
0 answers
30 views

Show that $\mathbb{Z}[i]=\{m+in\colon\; m,n\in\mathbb{Z}\}$ is a P.I.D [duplicate]

I came across the following problem Show that the ring of the Gaussian integers, defined as the subring of $\mathbb{C}$ given by the set $\mathbb{Z}[i]=\{m+in\colon\; m,n\in\mathbb{Z}\}$, where $i=\...
user avatar
3 votes
2 answers
123 views

What is the structure of $\mathbb Z[i]/\mathfrak p$ where $\mathfrak p$ is a prime.

Initially, I was trying to prove both the isomorphism of $\mathbb Z[i]/\mathfrak p\cong \mathbb{Z}[x]/(p,x^2+1)\cong \mathbb{F}_p[x]/(x^2+1)$, where $\mathfrak p$ is a prime in $\mathbb Z[i]$ for some ...
user avatar
  • 8,464
3 votes
1 answer
185 views

Quotient groups of the Gaussian integers of the form $\mathbb{Z}[i]/ c\mathbb{Z}[i]$

I was thinking about groups of the form $\mathbb{Z}[i] / c\mathbb{Z}[i]$ (which I will call $\mathbb{Z}[i]_c$ from now on) where $c$ is a Gaussian integer and addition is always the group operation. ...
user avatar
  • 1,760
3 votes
0 answers
219 views

Show that it is impossible to draw an equilateral triangle such that all three vertices are Gaussian integers.

Edit: I have seen the proof involving rational areas, but I came at this problem trying to use the roots of unity and rotations and translations of the complex plane. I couldn’t find a proof along ...
user avatar
  • 1,118
1 vote
2 answers
70 views

Minimal Polynomial of $1+i$ over $\Bbb Q$

What is minimal polynomial of $1+i$ over $\mathbb{Q}$? My attempt: Let $x=1+i$, hence $(x-1)^2= i^2 = -1$, which is not possible in $\mathbb{Q}$, hence squaring further we obtain the minimal ...
user avatar
  • 392
2 votes
2 answers
90 views

Power of Gaussian integer an integer

I'm trying to figure out when $(a+bi)^n$ is an integer, for integers $a,b,$ by putting certain conditions on $a$ and $b.$ I know that $b=0$ certainly works, but I'm not sure what other conditions ...
user avatar
1 vote
0 answers
45 views

Order in gaussian integer moduli

Say I want to find the order of an element in $\mathbb Z[i]$ modulo $a+bi.$ If this element is $x+yi,$ I could write $(a+bi)(c+di)+1=(x+yi)^n.$ I noticed that if $y=0,$ we could do this pretty easily ...
user avatar
-1 votes
1 answer
33 views

A question about Gaussian primes generated by the function $w(z)=z^2+i$ [closed]

While reading this paper and after some experiments, I asked myself the following question. Given the Gaussian integer $z_0=a+bi$, with $a\gt b$ and $b$ not divisible by $3$ its conjugate $\bar{z_0}=...
user avatar
1 vote
0 answers
133 views

Gaussian Integers and Finite Fields

I have been doing a little reading in elementary number theory, in particular I was reading about the Gaussian Integer, $\mathbb{Z}[i]$. My question regards the algebraic structure of $$\frac{\mathbb{...
user avatar
2 votes
0 answers
57 views

If $p$ is not sum of two squares then $p$ is a prime in $\mathbb{Z}[i]$

Let $p\in \mathbb{N}$ be a prime. Show that if $p$ is not the sum of two squares then $p$ is prime in $\mathbb{Z}[i]$ My approach was to assume $p$ is not a prime in $\mathbb{Z}[i]$. Then $\exists (s+...
user avatar
  • 1,404
1 vote
0 answers
42 views

Prime elements associate in $\mathbb{Z}[i]$

$p$ is prime and $p=a^2+b^2$ with $a,b\in \mathbb{Z}$. Show that $p=(a+ib)(a-ib)$ is a decomposition with prime elements in $\mathbb{Z}[i]$ and that $a+ib$ is associate with $a-ib$ if and only if $|a|=...
user avatar
  • 1,404
-1 votes
1 answer
57 views

The ring $\mathbb{Z}[i]/(2)$ [duplicate]

Let $\mathbb{Z}[i]/(2)$ be the ring of Gaussian Integers and $I=(2)$ be the ideal generated by $2$. Then $I=\{2(x+yi) | x,y \in \mathbb{Z} \}$. Therefore the elements of $\mathbb{Z}[i]/(2)$ are the ...
user avatar
  • 728
1 vote
1 answer
60 views

Cardinality of associate classes in quotient ring of Gaussian integers.

Let $n=up_1^{n_1}p_2^{n_2}\cdots p_k^{n_k}$, where $p_i$ is a prime and $u$ is a unit in $\mathbb{Z}[i]$ be the factorization of $n$ in $\mathbb{Z}[i]$. Then for any proper divisor $d$ of $\mathbb{Z}[...
user avatar
  • 572
7 votes
1 answer
142 views

Solving $x^2+100=y^3$ in $\mathbb{Z}^2$

Find all solutions to $x^2+100=y^3$ where $(x,y)\in\mathbb{Z}^2$. Here's my progress so far: Let $K=\mathbb{Q}(i)$, then $R:=\mathcal{O}_K=\mathbb{Z}[i]$ is a U.F.D. If $\alpha=x+10i$, then $\alpha\...
user avatar
5 votes
0 answers
234 views

Can you explain this relation between finite fields and circles?

Let $p$ be a prime such that $p \bmod 4 = 1$, so there exists some $i=\sqrt{-1}$ in $\mathbb{F}_p$. Furthermore, let $r \in \mathbb{N}$ be the radius of a circle such that there are $p-1$ lattice ...
user avatar
  • 245
8 votes
0 answers
191 views

Fermat's Last Theorem ($n=4$) using the Gaussian integers

I'm doing the second part of the following exercise in Miles Reid's Undergraduate Commutative Algebra: Exercise 0.18: Prove the cases $n=3$ and $n=4$ of Fermat's last theorem. I would like to know: ...
user avatar
2 votes
1 answer
44 views

Translation symmetry by Gaussian integers for this infinite product is impossible by Liouville. What takes its place?

On $\Bbb{R}$, we can construct the convergent infinite product $$F_{\Bbb{Z}}(x) := x \prod_{n \geq 1} \prod_{|k| = n} \left(1 - \frac{x}{k} \right) = x \prod_{n \geq 1} \left(1 - \frac{x^2}{n^2} \...
user avatar
0 votes
1 answer
31 views

Trivial Factor of Gaussian Primes

Hello anyone know how to prove that α=2+i is Gaussian primes? Here my step, so I let α=βγ then N(α)=N(β)N(γ), we get the norm of α which is N(α)=5 primes in Z. We get N(β)=1,N(γ)=5 or N(β)=5,N(γ)=1. ...
user avatar
2 votes
1 answer
76 views

Counting number of prime factorizations

The following is a problem from my elementary number theory class, which I believe the solution has an error. I'd appreciate if someone could verify that my hunch is correct, or help me understand why ...
user avatar
  • 1,423
2 votes
1 answer
72 views

(Proof explanation) $\mathbb{Z} /\left(a^{2}+b^{2}\right) \mathbb{Z} \cong \mathbb{Z}[i] /\langle a+b i\rangle$ doesn't hold

I was trying to solve the following exercise. I found the official solution to be too concise for me. If $a$ and $b$ are not relatively prime, then the $\operatorname{map} \mathbb{Z} /\left(a^{2}+b^{...
user avatar
  • 1,423
0 votes
0 answers
32 views

Example Gaussian Primes

I want to ask somethink. Is it possible to a number, prime in Z[i] but not prime in Z? If yes what's the example? If no, how to prove it?
user avatar

1
2 3 4 5 6