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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47 views

prove or disprove in $\mathbb{Z}[i]$ decomposition of real integer to 2 elements with the same norm is unique up to multiplying by unit [closed]

let $a,b,c\in\mathbb{Z}[i]$such that a=bc and say $|b|=B$ and $|c|=C$ . prove or disprove, there is no other decomposition such: $a=de ,|d|=B,|e|=C$ up to multiplying b and c by unity. Edit: this ...
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25 views

prove that $\forall k\geqslant1\land k\in\mathbb{Z}.\nexists q\in\mathbb{Z}.q^{2}=(k-i)(k+i)$

prove that $\forall k\geqslant1\land k\in\mathbb{Z}.\nexists q\in\mathbb{Z}.q^{2}=(k-i)(k+i)$. My try: use gaussian integer to claim that we have ufd. the problem, is that k might not be irreducible ...
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1answer
74 views

$1+i$ is a prime element of ring of Gaussian integers.

I want to prove that $1+i$ is a prime element of ring of Gaussian integers $Z[i]$. I started off with saying that if $1+i$ is prime then $1+i|(a+bi)(c+di)$ implies $1+i|a+bi$ or $1+i|c+di$ for $a+bi,c+...
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1answer
31 views

Any Gaussian prime dividing a product of gaussian primes is equal to one of those gaussian primes

I was asked to show that if any Gaussian prime $p$ divides the product $abc$ of Gaussian primes, then $p$ is equal to one of those primes, or one of them multiplied by a unit. I know that a Gaussian ...
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1answer
39 views

Find a gcd of two polynomials from $\mathbb{Z}[i]$

I'm trying to solve this problem from my abstract algebra course: Being $z_1=39-8i$ and $z_2=7+i$ elements from the ring of Gauss integers, $\mathbb{Z}[i]=\{a+bi \mid a,b\in\mathbb{Z}\}$. Find a ...
2
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1answer
52 views

$a+bi\mid n \implies a^2 + b^2\mid n$ if $(a,b)=1$ [duplicate]

This seems very simple, but I have not been able to come up with an answer to it. Suppose $a+bi \mid n$ in $\mathbb{Z}[i]$, where $a,b \in \mathbb{Z}$ with $(a,b)=1$. Prove that $a^2 + b^2 \mid n$ in ...
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3answers
77 views

Countability of $\mathbb{Z} \times \mathbb{Z}$ using isomorphism with $\Bbb Z[i]$ - how to make this rigorous?

Visualise the set $\mathbb{Z}\times \mathbb{Z}$ as points on the complex plane, and take origin as the center of a circle of radius $r$. With $r$ going from $0$ to $\infty$ - as you encounter points ...
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1answer
72 views

The gaussian integers modulo a pair of complex numbers

Taking a pair of Gaussian integers $u,v$ satisfying There exist $a,b \in \mathbb Z$ $v^2 = av+bu$ $u^2 = av+bu$ $uv = av+bu$ (Diffrent $a,b$) define the equivalence relation $\sim$ by $z\sim w \iff \...
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1answer
57 views

Given $N(\alpha) = pq$, $p$ ≡ $3$ mod $4$ and $q$ ≡ $3$ mod $4$, prove that $p = q$.

Suppose that $p,q \in \mathbb{N}$ are prime numbers and $\alpha \in \mathbb{Z[i]}$ satisfy $N(\alpha) = pq$, $p$ ≡ $3$ mod $4$ and $q$ ≡ $3$ mod $4$. Using the fact that $p$ and $q$ remain irreducible ...
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3answers
33 views

Find all solutions for $x \cdot y = 5$ in $\mathbb{Z}[i] $

Let $\mathbb{Z}[i] = \{a+ib \in \mathbb{C}, a, b \in \mathbb{Z} \}. $ Find all $x, y \in \mathbb{Z} [i] \backslash \{i, - i, 1,-1\} $ such that $$x \cdot y = 5.$$ I've denoted $x=a+ib, y=c+id $ and ...
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25 views

Gaussian rationals and Gaussian integers

The Gaussian integers are defined as the numbers that can be written as $a + bi$ with $a, b$ rational numbers, and for which there is a monic polynomial $P \in \mathbb{Z}[X]$ such that $P(a + bi) = 0$....
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38 views

Finding complete residue systems in Gaussian integers

Let $\mathbb{Z}[i]$ be the ring of Gaussian integers. A set $C\subset \mathbb{Z}[i]$ is called a complete residue system (CRS) modulo $\alpha$ if for every Gaussian integer $\pi$ there exists a unique ...
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1answer
71 views

Indication of Gaussian integers solutions for $x^2+y^2=z^3$.

I know the existence of infinitely many integer solutions for $x^2+y^2=z^3$. But my concern is the existence of Gaussian Integers solution of this diophantine equation. I try to search for relevant ...
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48 views

How to prove that $\mathbb{Z}[\exp(\frac{2\pi i}{3})]$ is a Euclidean domain?

To give the context I am currently studying Gaussian Integers, and I have of course studied rings. The full question is: Given $\rho = \exp(\frac{2\pi i}{3})$, show that the ring $R = \mathbb{Z}[\rho]$...
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2answers
54 views

Can't understand part of proof of Catalan conjecture

So I'm trying to understand this proof or at least part of it and I'm stumped at a point that seems pretty basic. In the ring $Z[i]$ we have $x^p = y^2 + 1 = (y - i)(y + i)$. It is known that $Z[i]$ ...
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26 views

What are the minimal natural coefficients allowed for $a+bi$ in the residue class ring modulo $z$, where $z$ is a none-integer prime?

I apologize for the messy title. Say I have an integer $p$ which is prime in $\Bbb Z$, and is of the form 4k+1. Over the gaussian integer ring, $p$ isn't prime and there exists a prime, none-integer $...
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1answer
57 views

Multiplicative inverse of a gaussian integer

So, I had to prove that Gaussian integers had an identity element and ended up with it being $(1 + 0i)$. Now I have to see if any $(a+bi)$ (except $(0 + 0i)$)has a multiplicative inverse. Then I ended ...
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68 views

$\mathbb{Z}/\left(a^{2}+b^{2}\right) \mathbb{Z} \cong \mathbb{Z}[i] /\langle a+b i\rangle$ (Solution verification)

The following is a proposed solution to a fairly routine problem (which has in fact been asked on stackexchange Quotient ring of Gaussian integers, but not as a solution verification question), which ...
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2answers
46 views

How to show that $2+\sqrt{5}i\not | 3$? [duplicate]

How can I prove that there is no $x\in \mathbb{Z}[\sqrt{-5}]$ such that $x\cdot (2+\sqrt{5}i)= 3$ ?
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241 views

Showing that an ideal is equal to $\mathbb{Z}[i]$

Let $a$, $b$, and $c$ be three elements of $\mathbb{Z}[i]$, the ring of Gaussian integers. We define the following to be subset of $\mathbb{Z}[i]$: $$(a,b,c)=\{ax+by+cz:\ x,y,z \in\mathbb{Z}[i]\}.$$ ...
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27 views

How do I find $\displaystyle {\inf\limits_{n,m} \dfrac {\deg (m)} {\deg (r)}}\ ?$

Given $n,m \in \Bbb Z[i],$ we have $n = qm + r,\ r \in \Bbb Z[i].$ Then what is the value of $\displaystyle {\inf\limits_{n,m} \dfrac {\deg (m)} {\deg (r)}}\ ?$ How do I solve this question? Any help ...
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1answer
31 views

Does the non-zero ideal $I=\langle a+ib\rangle$ contain no positive integers?

I have a question as follows given by my professor: Does the non-zero ideal $I=\langle a+bi\rangle$ contain no positive integers? I answered as follows: Since, $I$ is non-zero so, $a+bi$ is non-zero ...
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1answer
32 views

Why cannot this Gaussian integer be expressed as a sum of squares?

Gaussian integers are complex numbers whose real and imaginary parts are both integers. Give the Gaussian Integer a=8-53i, show that it cannot be expressed in the form $w^2+z^2$ where w and z are GI's....
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1answer
21 views

Which of these integers can be written as the sum of two squares of integers? [duplicate]

I know that an integer is the sum of two squares if and only if it is the norm of some Gaussian integer. However, the numbers below are so large I don't even know where to begin, let alone know ...
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18 views

Describe the elements of $\mathbb{Z}[i]/(3)$ [duplicate]

I would like to describe the elements of $\mathbb{Z}[i]/(3)$. So I know that these elements look like $z+3\mathbb{Z}[i]$ and I tried to see when we have that $\hat{x}=\hat{y}$. I saw that this happens ...
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56 views

Modular inverse by Euler's theorem for Gaussian integers

To compute the multiplicative inverse modulo a prime one can use extended GCD or Euler's theorem: $$x^{-1} = x^{p-2} \mod p$$ Is there similar formula for x being a Gaussian integer? I'm asking ...
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55 views

Show that every ideal in the ring of Gaussian integers is principal

I'm doing Exercise 9 in textbook Algebra by Saunders MacLane and Garrett Birkhoff. Show that every ideal in the ring of Gaussian integers is principal. Could you please verify if my attempt is fine ...
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78 views

Proof of Euclidean division algorithm for the ring of Gaussian integers

I'm trying to prove this property from this Wikipedia page. Let $\mathbb Z[i]$ denote the ring of Gaussian integers and $N(z) = z \overline{z}$ for $z \in \mathbb Z[i]$. For $p,q \in \mathbb Z[i]$ ...
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1answer
39 views

Euler totient function for Gaussian integers

Actually this not a question but just an observation followed by a small question. For primes $p$, $\phi(p)=p-1$, for general $n$, $\phi(n)<n$ since $\phi(n)$ is the order of the group $\mathbb{Z}...
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2answers
110 views

Proving that a Gaussian integer $a+bi$ with $a,b\neq0$ is composite in $\mathbb{Z}[i]$ if its norm $a^2+b^2$ is composite in $\mathbb{N}$?

It is simple to show for any given composite sum of two squares, that there is at least one Gaussian composite with that norm, but more difficult to show that all Gaussian integers with that norm are ...
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2answers
50 views

Prove $(7)$ is maximal in Gaussian Integers

I would like to prove that $(7)$ is a maximal principal ideal in Gaussian Integers. I approached this problem by trying to prove that $7$ is prime in $\Bbb Z[i]$. However, I am not sure how to proceed ...
3
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0answers
72 views

Is there a known necessary and sufficient criterion for a Gaussian Integer $a+ib$ to be prime in $\mathbb{Z}[i]$? [duplicate]

I am a high school student interested in studying mathematics at university. I've been doing some independent study and have found the Gaussian Integers $\mathbb{Z}[i]$ particularly interesting. I was ...
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0answers
59 views

Gaussian Prime must have size $p$ or $p^2$ [duplicate]

Chapter 38 of Mazur and Stein's "Prime Numbers and the Riemann Hypothesis" asks us to prove that if a Gaussian integer is a prime Gaussian integer, then its norm can only be an "...
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2answers
131 views

What are the unit elements in $\Bbb{Z}[i]$? [duplicate]

What are the unit elements in $\Bbb{Z}[i]$, where $\Bbb{Z}[i]$ is defined to be the set of Gaussian integers ? My progress: My mentor gave this problem and he said to use determinants and scalefactor. ...
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0answers
30 views

Showing the property that norm in the ring $\mathbb{Z}[\delta]$ with $\delta=\frac{1+\sqrt{d}}{2}$, $d<0$ is multiplicative.

Following my other thread: Finding units of $\mathbb{Z}[\delta]$ with $\delta=\frac{1+\sqrt{d}}{2}$ and $d\equiv 1\pmod{4}.$ In the ring $\mathbb{Z}[\delta]:=\left\{a+b\delta:a,b\in\mathbb{Z}\right\}$ ...
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3answers
57 views

If $(a+bi)$ is a unit in $\mathbb{Z} [i]$, then $N(a+bi) = 1$?

How does one prove that, if some $a+bi \in \mathbb{Z}[i]$ is a unit, then the norm of $a+bi$, $N(a+bi) = 1$ without simply checking each unit? I've tried a few different things (applying the Well-...
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2answers
173 views

How many unit squares of a square grid overlap a circle of given radius centered on the origin?

Consider, in the plane, the unit squares with corners having integral rectangular coordinates. Let $N_r$ be the number of these unit squares whose interior is intersected by a circle of radius $r$ ...
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0answers
24 views

Characterization of Gaussian numbers, proof explanation

$\mathbb{Z}[i]$ consists of exactly the numbers of the field extension $\mathbb{Q}(i)|\mathbb{Q}$, which fulfull the normed equation $x^2+ax+b=0$ with coefficients $a,b\in\mathbb{Z}$. Proof: An ...
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2answers
43 views

Enumerate the elements of a quotient Gaussian Integers ring

I want to find an extensive list of all the elements of the quotient ring $Z[i]/(3+i)$. Since the Gaussian integers are an euclidean domain with euclidean function $N(a+bi)=a^2+b^2$ the representative ...
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43 views

“vector space”-like set over non-field

The definition of vector space is always defined over a field. Even for finite fields (I believe, for example $\mathbb{Z}\mathrm{mod}p$). Integers are not a field because there is no general inverse ...
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1answer
101 views

Why does the characterization of Gaussian primes really work?

Citing https://en.wikipedia.org/wiki/Gaussian_integer : A Gaussian integer $a + bi$ is a Gaussian prime if and only if either: one of $a, b$ is zero and absolute value of the other is a prime number ...
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2answers
38 views

GCD of Gaussian Integers $\text{gcd}(4, 36+18i)$

I have to compute $\text{gcd}(4, 36+18i)$. I computed the norms: $16$ and $1620$. I am sure $2$ is the gcd. Is there any method to prove $2$ is the gcd, other than using the Euclidean Algorithm (...
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0answers
32 views

Prove an infinite product factorization of lemniscate sine

I recently read about the lemniscate sine function. The function $sl$ is defined as the inverse of $\mathrm{arcsl}(x)=\int_0^x \frac{\mathrm{d}t}{\sqrt{1-t^4}}$. We know that it is an elliptic ...
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1answer
69 views

Use the Euclidean algorithm to find the GCD of $1 + 28i$ and $4 + 7i$

I am trying to learn how to use the Euclidean algorithm to find the GCD of $1 + 28i$ and $4 + 7i$. The question I am trying to answer is: Apply the Euclidean algorithm to $\alpha = 1 + 28i$ and $\...
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1answer
49 views

If p (mod 4) = 3 and p is a Gaussian Prime. How to show that Z[i]/(p) is equal to GF(p^2)/(x^2+1)?

I currently work with Gaussian Integer. I try to use prime Gaussian Integer field for Elliptic Curve instead of prime field. . We know that every finite field isomorphic to polynomial field with ...
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2answers
64 views

What is a generator for an ideal such that $I=\{a+bi|a+b \text{ is even}\}$?

I had this problem where i had the application $\varphi: \mathbb Z[i] \Rightarrow \mathbb Z/(2)$ where $\varphi(a+bi)=\bar{a}+\bar{b}$. I had to find the kernel and prove that is a factor ideal. I ...
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2answers
74 views

Number Theory: Quadratic Fields

Suppose $32 =\alpha \beta $ for $\alpha,\beta $ relatively prime quadratic integers in Q[i]. Show that $\alpha= \varepsilon \gamma^2 $ some unit $\varepsilon $ and some quadratic integer $ \gamma $ ...
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0answers
45 views

For $p$ prime in $\mathbb{Z}[i]$, show $p \equiv 3 \mod 4$ using ring theory. [duplicate]

Just building on a previous question, where we showed that an odd number $p$ is prime in $\mathbb{Z}[i]$ if and only if $x^2+1$ does not have roots in $\mathbb{Z}/p\mathbb{Z}$. The last part of this ...
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2answers
50 views

For $p$ odd, show that $p$ is prime in $\mathbb{Z}[i] \iff x^2+1$ does not have roots in $\mathbb{Z}/p\mathbb{Z}$

I am building on a previous question, where we showed that $\mathbb{Z}[x]/(x^2+1) \cong \mathbb{Z}[i]$. Now, I want to prove the statement in the title of this question. First, let $p$ be an odd ...
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41 views

What are the zero divisors of the ring $\mathbb Z_n[i]$, where $n$ has only one prime factor which is congruent to 3 (mod 4)?

What are the zero divisors of the ring $\mathbb Z_n[i]$, where $n$ has only one prime factor which is congruent to 3 (mod 4)? I feel like this is the set of all those elements where real and ...

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