# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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### 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-...
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### 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 ...
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### 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. \{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 }\} ...
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### 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. \{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 }\} ...
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### 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 $p|(a^2+1)$ ...
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### 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) = (...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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} \...
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### 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. ...
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### 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 ...
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### (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^{...