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Questions tagged [gaussian-integers]

A Gaussian integer is a complex number whose real and imaginary parts are both integers.

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How to prove $\Bbb Z[i]/(1+2i)\cong \Bbb Z_5$? [duplicate]

How to prove $\Bbb Z[i]/(1+2i)\cong \Bbb Z_5$? My method is $$(1+2i)=\big\{a+bi丨a+2b≡0\pmod 5\big\},$$ So any $a+bi$ in $\Bbb Z(i)$,we got $$a+bi=(b-2a)i+a(1+2i).$$ So $\Bbb Z[i]/(1+2i)=\big\{0,[...
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Gaussian Normalization Method

Kindly I have searched for Gaussian Normalizing method but could not find it. Please, could you guide me? what is the equation, and how it is work? Thank you
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Roots of polynomials are Gaussian integers

I have got a question. I want to show the following: Let P be a normalized polynomial with integer coefficients and let w be a root of this polynomial (in $\mathbb{Q}[i]$), then w is a Gaussian ...
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1answer
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Solving the Diophantine equation $y^3 = 4x^2+4x+ 5$ for $x,y \in \mathbb{Z}$

I want to solve the Diophantine equation $y^3 = 4x^2+4x+ 5$ for $x,y \in \mathbb{Z}$. The right hand side factors as $(2x+1-2i)(2x+1+2i)$. Am I right that such a factorization can be found using the ...
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1answer
32 views

Homomorphic images of Gaussian Integers

Let $x=a+bi$ be an arbitrary Gaussian integer and consider the qoutient ring $S := \frac{\mathbb{Z}[i]}{(x)}$. I know that the number of elements of $S$ is equal to $a^2 + b^2$. Is it true that $S$ is ...
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1answer
18 views

Describing Cosets in $R/A$

In a worked example in my textbook, we are describing the cosets in $R/A$, where $R=\mathbb{Z}[i]$, the Gaussian integers, and $A = (2+i)R$, the ideal of all multiples of $2 + i$. It starts by ...
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2answers
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The number of surjective ring homomorphism from $\mathbb{Z}[i]$ to $\mathbb{F}_{11^2}$.

Find the number of surjective ring homomorphism from $\mathbb{Z}[i]$ onto $\mathbb{F}_{11^2}$. If such a surjective ring homomorphism exists with kernel $(a+bi)$, then $\mathbb{Z}[i]/(a+bi)\cong\...
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0answers
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Showing that $\mathbb Z[i]/(p)\simeq \mathbb Z[i]/(\pi)\times\mathbb Z[i]/(\overline\pi )$

Let $p\in \mathbb Z$ be a prime congruent to $1\mod 4$. Let $p=\pi\overline \pi$. Show that $\mathbb Z[i]/(p)\simeq \mathbb Z[i]/(\pi)\times\mathbb Z[i]/(\overline\pi )$. Show that the quotient ring $\...
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2answers
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Write $5$ and $6+8i$ as products of irreducible elements in $\mathbb{Z}[i]$.

Write $5$ and $6+8i$ as products of irreducible elements in $\mathbb{Z}[i]$. I am not sure how to do $5$. Do I assume $5=\alpha \beta$, and so $v(\alpha\beta)=v(\alpha)v(\beta)=5$. Then since $5\...
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1answer
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Find number of invertible elements in $\mathbb{Z}[i]/(220+55i)\mathbb{Z}[i]$

I was able to find the factorization $220+55i=11*(2+i)*(2-i)*(4+i)$. Also know this famous question Quotient ring of Gaussian integers But how to apply it in this case? I'm confused, please help. ...
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1answer
21 views

The intersection of set of multiples of a G/E integer and the set of integer

Suppose that q is a Gaussian or Eisenstein prime and let p be the prime number that lies below q. S is the set of G/E multiples of q. How to prove that S∩Z is the set of integer multiples of p? I'm ...
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1answer
129 views

Need help with Euclidean Algorithm in $\mathbb{Z}[i]$

I'm trying to find the GCD of $(85,1+13i)$ and $(47-13i,53+56i)$. I've tried, but to no avail. I keep setting it up and trying to do it with the same mindset as if i'm doing polynomial division, is ...
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1answer
335 views

Prove $(5-i,13)$ is a principal ideal in $\mathbb{Z}[i]$

I'm doing the same exercise as the one asked about in this post. The only part I was not able to solve (the rest of the exercise is not relevant for this part) is to prove that $(5-i,13)$ is a ...
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1answer
26 views

A way of dividing in $\mathbb{Z}[i]$

I believe I have an algorithm for division in $\mathbb{Z}[i],$ but I can't seem to prove it works, nor can I find references for it online (although my searches for "division algorithm in Z[i]" seem ...
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Factorization of the ring $\mathbb{Z} [i]$ of Gaussian Integers

Prove that if a pair of integers a and b is a solution to the equation $$x^2 +y^2=n$$ Then $n$ factors $\mathbb{Z} [i]$ as the product $(a+bi)(a-bi)$ I understand that in $\mathbb{Z} [i]$ that $a^2 +...
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2answers
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Modular inverse of Gaussian Integers

Let $f_0$ and $f_1$ be Gaussian Integers such that $f_0 = a + i$ and $f_1 = b + i$. How can I compute $f_0^{-1} mod f_1$ and $f_1^{-1} mod f_0$? I've been trying to apply the Extended Euclidean ...
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3answers
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Where is wrong with this fake proof that Gaussian integer is a field?

The Gaussian integer $\mathbb{Z}[i]$ is an Euclidean domain that is not a field, since there is no inverse of $2$. So, where is wrong with the following proof? Fake proof First, note that $\...
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“up to associates” in the Euclidean domain

We know that the Euclidean Domain has the property of Unique Factorization. More precisely, every nonzero element in a Euclidean ring $R$ can be uniquely written (up to associates) as a product of ...
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Show that if $\beta \mid \alpha$ in $\mathbb Z[i]$ then $N(\beta) \mid N(\alpha)$

Show that if $\beta \mid \alpha$ in $\mathbb Z[i]$ then $N(\beta) \hspace{1mm}| \hspace{1mm} N(\alpha)$ where $\alpha$ is a prime in $\mathbb Z[i]$ and $N(a + bi) = a^2 + b^2$. So we are working ...
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0answers
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Finding an explicit map associated with the quotient ring of Gaussian integers [duplicate]

Let $a+bi\in\mathbb{Z}[i]$ with $\gcd(a,b)=1$. I know that $\mathbb{Z}[i]/\langle a+bi\rangle\cong\mathbb{Z}_{a^{2}+b^{2}}$ by a ring homomorphism $\phi:\mathbb{Z}[i]\to\mathbb{Z}_{a^{2}+b^{2}}$, ...
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1answer
129 views

Parameterization of Equation with Gaussian Integer

I'd like to ask how to get the parameterization to this equation: $3z_1^2+z_2^2=156$, where $z_1$ and $z_2$ are both Gaussian integers. More generally, is there any parameterization to the general ...
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1answer
77 views

How to count the number of elements in $(\mathbb{Z}[i]/I^{2014})\otimes_{\mathbb{Z}[i]}(\mathbb{Z}[i]/J^{2014})$?

Let $I,J\unlhd \mathbb{Z}[i]$ be the principal ideals generated by $7-i$ and $6i-7$, respectively. Find the number of elements in the $\mathbb{Z}[i]$-module $A=(\mathbb{Z}[i]/I^{2014})\otimes_{\mathbb{...
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2answers
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How to show that the kernel of this ring homomorphism $\phi: \mathbb{Z}[i]\to\mathbb{Z}/5\mathbb{Z}$ is the principal ideal $(2+i)$?

In order to prove that $Z[i]/(2+i)\cong \mathbb{Z}/5\mathbb{Z}$, I defined the homomorphism $\phi(a+bi)=(a-2b)\mod 5$. It is easy to show that $2+i\in\ker(\phi)$, but I'm unsure how to prove the ...
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0answers
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Square and cubic root of gaussian Integer

I am looking for an algorithm to calculate the Square-root (cube-root?)of an Gaussian integer n, where n isn't a square number. So i define the integer square root of an gaussian-integer n as the ...
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1answer
34 views

Value for sum of inverse of Gaussian integers

So I was studying some stuff about lattices and at some point I reached the Eisenstein series of weight 2k given by $$G_{2k}(\Lambda)=\sum_{w\in\Lambda-\{0\}}w^{-2k}$$ Then I tried to think about "...
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Exercise about primes in the ring of Gaussian integers

Let $p$ be a prime in $\mathbb{Z}$ of the form $4n + 1, n \in \mathbb{N}$. Show that $\left(\frac{-1}{p}\right) = 1$ (here $\left(\frac{\#}{p}\right)$ is the Legendre symbol). Hence prove that $p$ is ...
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0answers
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Creating realization of 2D Gaussian field in Fourier space

I want to generate a 3D Gaussian field with dimensions $L\times L\times L$ with $N^3$ cells each of size $l = L/N$. I'm doing this by producing a realization of this field in Fourier space by ...
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2answers
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Product of $(4k-1)$ primes can't be sum of 2 squares

I am trying to prove, Product of primes of the form $(4k-1)$ can't be sum of 2 squares. My approach is- Let the product is $M=m_1m_2...m_n$ where $ m_1, m_2, ...m_n$ are primes. Assume, $M$ can ...
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Let $\alpha \in \mathbb Z [\sqrt{-1}]$. Is it true that $\mathbb Z[\alpha]/I$ is finite for any non-zero ideal $I$ of $\mathbb Z[\alpha]$?

Let $\alpha \in \mathbb Z [\sqrt{-1}]$. Is it true that $\mathbb Z[\alpha]/I$ is finite for any non-zero ideal $I$ of $\mathbb Z[\alpha]$? I ask because on an old exam paper it asks me to prove this ...
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number theory problem, about arctan and pi

I have encountered a problem, that is: Find all integers $a,b,c,d$, so that $$ \frac{\pi}{4}=a\arctan\left(\frac1b\right)+c\arctan\left(\frac1d\right). $$ Is it possible to find all solutions ...
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2answers
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Nonunit element in $\mathbb{Z}[i]$

If $a+bi$ is not a unit of $\mathbb{Z}[i]$ prove that $a^2+b^2>1$. Definition: Let $R$ be commutative ring with unit element. An element $u\in R$ we call unit if it's inverse $u^{-1}$ also lies in ...
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When does a $n \in \mathbb{Z}$ can be written as $n=A^2+B^2$?

I want to establish the condition which will determine when does an element $~n \in \mathbb{Z}$ can be written as two integer squares i.e., $~n=A^2+B^2$ for $A, B \in \mathbb{Z}$. Now I have found the ...
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1answer
146 views

Ring of Gaussian integers modulo p=4n+3

How can I prove that there are no Zero divisors in the Ring of Gaussian integers modulo p=4n+3, where p is prime, n is integer? Thank you.
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Solve Equations in the ring of Gaussian Integers

How to solve equations in Gaussian integers? For example: $$(7-i)x+(12-i)y = 2+3i.$$ Why can't I just rewrite it as $7x+12y = 2; -x-y = 3$? However, in that case the solution isn't integer.
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If $ p \equiv 3 (\bmod{4}) $ is a rational prime, then p is a Gaussian prime

I understand the proof that my professor gave in class for the most part, but there's one snag that I'm having. Here's the proof to begin with [Side note: $N(x)=a^2+b^2$, where $a+bi$ is a complex ...
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2answers
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Question about a proof of $\mathbb{Z}[i]/(2+i)\cong \mathbb{Z}/5\mathbb{Z}$

Proof: Since $\mathbb{Z}[i]\cong \mathbb{Z}[x]/(x^2+1)$, where $i$ and $x$ correspond, we have: $\mathbb{Z}[i]/(2+i)\cong \mathbb{Z}[x]/(x^2+1,2+x)$. In that ring, since $x=-2$, $x^2+1=...
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1answer
42 views

Let $α=a+bi \in\mathbb{Z}[i]$ with $\gcd(a, b)=1$. Show that there exists $c\in\mathbb{Z}$ such that $c+i$ is a multiple of $α$ in $\mathbb{Z}[i]$.

Let $α=a+bi \in\mathbb{Z}[i]$ with $\gcd(a, b)=1$. Show that there exists $c\in\mathbb{Z}$ such that $c+i$ is a multiple of $α$ in $\mathbb{Z}[i]$ (that is, $c + i = αδ$ for some $δ \in\mathbb{Z}[i]$)....
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3answers
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Prove that $\mathbb{Z}[i]$ is an integral domain.

Could someone please verify whether my solution is okay? Prove that $\mathbb{Z}[i]$ is an integral domain. Claim: $\Bbb{Z}[i]$ is a commutative ring. Let $a+bi,c+di\in \Bbb{Z}[i]$. Then $(a+bi)(c+...
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2answers
39 views

If the norm of $\alpha \in \mathbb{Z}[i]$ is a square, show that $\alpha$ is a square in $\mathbb{Z}[i]$

Suppose $\alpha \in \mathbb{Z}[i]$, $\alpha$ is not divisible by any integer, and $N(\alpha) = m^2, m \in \mathbb{Z}$. I want to show that $\alpha$ is a square in $\mathbb{Z}[i]$. I'm really lost ...
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Solve in $\mathbb{Z}$ the equation $x^4 + 1 = 2y^2$.

Find all pairs of intergers $(x,y)$ such that $x^4 + 1 = 2y^2$. I'm thinking of Gaussian integers, since the LHS can be factored in $\mathbb{C}$. But I don't know how to continue here.
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1answer
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Are there any interesting results in quadratic extensions that adjoin $2^k$th roots of unity beyond the Gaussian Integers?

This is admittedly a bit of a broad question I realize, but curiosity has struck me a bit lately. So before you tl;dr; here's the central question: Are there any known generalizable results that arise ...
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1answer
57 views

Deducing whether elements are prime in $\mathbb{Z}[\sqrt{-2}]$

Let $ \alpha_1 = 3 + 2\sqrt{-2}, \alpha_2 = 1 + 2\sqrt{-2}$, how do I deduce if these elements are prime in $\mathbb{Z}[\sqrt{-2}]$? I've tried showing that they aren't irreducible, which in turn ...
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0answers
44 views

$Z[i]/(2+3i) \simeq Z/13Z$ is my proof correct? [duplicate]

Here's my new attempt at showing $Z[i]/(2+3i) \simeq Z/13Z$. proof: Let's define the natural homomorphism $$\phi: Z \to Z[i]/(2+3i)$$ where $$\phi(z)= z+ (2+3i)$$ It is easy to check that this is a ...
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3answers
83 views

Let $I=(2,1+\sqrt{-5}), J=(3,1-\sqrt{-5}) \trianglelefteq \mathbb{Z}[\sqrt{-5}]$, show $IJ=(1-\sqrt{-5})$

Let $R=\mathbb{Z}[\sqrt{-5}]$ What is $R^\times$ Let $I=(2,1+\sqrt{-5})$, is $I$ a principal ideal in $R$? Let $J=(3,1-\sqrt{-5})$, prove $I+J=R$ Prove $IJ=(1-\sqrt{-5})$. I was able ...
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3answers
360 views

Reason why in Gaussian integers, norm divisibility may not lead to divisibility.

It is taken as true (with a very easy proof) for Gaussian integers, that for $\alpha, \beta \in \mathbb {Z}[i]$, if $\beta \mid \alpha$ then $N_{\beta} \mid N_\alpha$ in $\mathbb {Z}$. It would be an ...
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1answer
41 views

Significance of non-unique quotient and remainder for Gaussian integers.

In the context of Gaussian integers, there can be non-unique quotients and their corresponding remainder, as for the example at page 6 of this, that states example of dividend($a$) of $3+2i$, divisor($...
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2answers
54 views

Finding values of quotient and remainders for Gaussian integers.

There is an example copied from example at page 6 of this, that states example of dividend of $3+2i$, divisor of $-1+3i$, and states that as long as the $N_r \lt N_b$, (with $r$ denoting remainder, $b$...
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0answers
165 views

Proving Gaussian integers have non unique quotient and remainder in division algorithm.

As given in the book by Pollard, Diamond, on Algebraic Numbers, there is proof for division algorithm being made to work for Gaussian integers, with the above as an exercise left to the reader to ...
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2answers
429 views

How to prove that 2 is not a Gaussian prime?

To prove an algebraic number is not a Gaussian prime, need find factors of it. Let us assume, $(a + bi)(x +yi) = 2, \exists a,b,x,y \in \mathbb{Z}$; so $(ax -by) + i(ay + bx) =2$. As imaginary part ...
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0answers
38 views

Is my reasoning classifying gaussian primes right?

what I want to prove: if $a+bi$ (both a and b are nonzero)is gaussian prime,then $N(a+bi)$ is prime integer. I know the fact that if $a+bi$ is gaussian prime ,then its norm must have the form $p^2$ ...