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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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Seeking elementary proof: For prime $p$ such that $4\mid p -3$, the exponent of $p$ in the prime decomposition of $m^2+n^2$ is even

The theorem I am referring to is the following: If $m$ and $n$ are integers, and $p$ is a prime number such that $4 \mid p - 3$, then the exponent of $p$ in the prime decomposition of $m^2 + n^2$ ...
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1answer
23 views

Describing the elements of quotient ring of $\mathbb{Z}[\sqrt{D}]$.

For Gaussian integer ring $\mathbb{Z}[i]$, there is a method describing distinct elements of certain quotient ring of $\mathbb{Z}[i]$ using 'the visualization'. The images(in K. Conrad's note) below ...
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0answers
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Counting $2\times2$ Orthogonal matrices over the ring $\Bbb{Z}_p[i]$. [duplicate]

Our research is about counting the number of orthogonal matrices over the ring of Gaussian integers modulo $p$. A matrix $A$ is said to be orthogonal if $AA^T=I$. My question is how many $2\times2$ ...
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35 views

Find the values of $a\in \mathbb{Z}[i]$ such that $(2,1)$ and $(2+i,a)$ form a basis of $\mathbb{Z}[i]^2$.

I'm trying to solve an exercise which asks me to determine for what values of $a\in \mathbb{Z}[i]$ $(2,1)$ and $(2+i,a)$ form a basis of $\mathbb{Z}[i]^2$ (where we're considering $\mathbb{Z}[i]^2$ as ...
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1answer
629 views

How much of an infinite board can a N-mover reach?

This question is inspired by the question on codegolf.SE: N-movers: How much of the infinite board can I reach? A N-mover is a knight-like piece that can move to any square that has a Euclidean ...
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1answer
35 views

Show that $ | \mathbb Z[\sqrt{-1}]/(a+b\sqrt{-1}) |=a^2+b^2 $ [duplicate]

Let $ a+b\sqrt{-1} $ be a non-zero element of the ring of Gaussian integers $ \mathbb Z[\sqrt{-1}]. $ Show that $ | \mathbb Z[\sqrt{-1}]/(a+b\sqrt{-1}) |=a^2+b^2 $. [Jacobson, Basic Algebra, P202.3] ...
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4answers
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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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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
89 views

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
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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
19 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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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
78 views

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
23 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
130 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
380 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
110 views

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
2k views

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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3answers
184 views

“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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3answers
44 views

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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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
151 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
79 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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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
63 views

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
40 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
24 views

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
119 views

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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2answers
68 views

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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0answers
61 views

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
80 views

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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0answers
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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
171 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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3answers
76 views

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
946 views

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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4answers
276 views

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
78 views

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
63 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
47 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
90 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
402 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 ...