# Questions tagged [gaussian-integers]

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

156 questions
48 views

### 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 $m^2 + n^2$ ...
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 ...
17 views

### 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 $2\times2$ ...
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 ...
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 ...
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] ...
125 views

40 views

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. ...
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 ...
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 ...
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 ...
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 ...
40 views

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 ...
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 ...
77 views

### 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 $\phi:\mathbb{Z}[i]\to\mathbb{Z}_{a^{2}+b^{2}}$, ...
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 ...
79 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 $δ \in\mathbb{Z}[i]$)....