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Questions tagged [eisenstein-integers]

For questions about Eisenstein integers. The Eisenstein integers are the complex numbers of the form $a+b\omega ,$ where $\omega=e^{2\pi i/3} $ is a primitive $3$rd root of unity. Sometimes they're also called the Eulerian integers. They form a subring of $\Bbb Q(\omega).$

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Is $c = \frac{\sqrt 3}{4} \frac{\pi}{4} \prod_{p = 2 \mod 3} \sqrt{\frac{p^2}{p^2-1}} \prod_{q = u^2 + 3 v^2} \sqrt{\frac{q^2}{q^2-1}}$?

Consider the sum of $2$ squares and Gauss circle problem https://en.wikipedia.org/wiki/Gauss_circle_problem and also The Landau-Ramanujan Constant that relates to the sum of 2 squares. See : http://en....
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Is every sufficiently large gaussian integer the sum of 3 cubes ? $a + b i = (c + di)^3 + (e + fi)^3 + (g + hi)^3$?

Is every sufficiently large gaussian integer the sum of $3$ gaussian cubes ? In other words, $$a + b i = (c + di)^3 + (e + fi)^3 + (g + hi)^3$$ for given integers $a,b$ with $a^2 + b^2 > Q$ can ...
mick's user avatar
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Law of cubic reciprocity for a primary prime of norm 3

I am currently studying "A Classical Introduction to Modern Number Theory", a book by Kenneth Ireland and Michael Rosen. In that, we have the following two theorems: $(1)$ The Law of Cubic ...
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Cubic non-residue calculation

I am currently studying Cubic residue characters from Kenneth Ireland and Michael Rosen's "A Classical Introduction to Modern Number Theory", and this is the definition given in the book: If ...
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Eisenstein integers with norm value of 5

I am trying to find out the Eisenstein integers that have norm values of 3,5,7.... I want to see if there is some pattern. For the norm value of 3, I was able to find six Eisenstein integers. For the ...
Disha's user avatar
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Let $\pi \in \mathbb{Z}[\omega]$ be a prime. Then $\frac{\mathbb{Z}[\omega]}{\pi \mathbb{Z}[\omega]}$ has $N(\pi)$ elements

I am currently reading cubic and biquadratic reciprocity from Kenneth Ireland and Michael Rosen's, "A Classical Introduction to Modern Number Theory" have a doubt in proposition 9.2.1. We ...
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Automorphism groups of $\mathbf Z$, $\mathbf Z[i]$, $\mathbf Z[\omega]$

I would like to know the automorphism groups of the rational integers $\mathbf Z$, the Gaussian integers $\mathbf Z[i]$, and the Eisenstein integers $\mathbf Z[\omega]$. My question is, would $\text{...
node196884's user avatar
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$\exists X, Y, Z \in \mathbb{Z}[\omega]$ such that $X^3 + Y^3 + Z^3 = \omega$?

I am considering the following problem: Denote by $\mathbb{Z}[\omega]$ the set of Eisenstein integers. Let $X, Y, Z \in \mathbb{Z}[\omega]$ be non-zero integers coprime to $1-\omega$. Is it possible ...
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Eisenstein Integers modulo $(1-\omega)^2$

I wish to find the addition and multiplication tables of Eisenstein integers modulo $(1-\omega)^2 = -3\omega$. In drawing the fundamental parallelogram with vertices at $0, z = - 3\omega, z\omega = 3+...
V. Elizabeth's user avatar
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For Eisenstein numbers $a + b\frac{1+\sqrt{3} i}{2}$ with $0 \le a, b \le n$ how can I efficiently identify all of the primes?

This answer to determining if a coincident point in a pair of rotated hexagonal lattices is closest to the origin? teaches that my real question was how to find if two Eisenstein integers are coprime. ...
uhoh's user avatar
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If $x \equiv 2 \pmod 3$, show that $x=\pm x_1 x_2 \cdots x_t$, where $x_i\equiv 2 \pmod 3$

This question comes from A Classical Introduction to Modern NT (Ireland & Rosen), chapter $9$ $\#17$. Note that $D$ is the set of Eisenstein integers. Exercise 9.17 An element $x \in D$ is called ...
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$p\equiv 2\pmod 3$ is an odd prime. Prove that there are no integers $x$, $y$ satisfying $p=x^2-xy+y^2$. [duplicate]

$p\equiv 2\pmod 3$ is an odd prime. Prove that there are no integers $x$, $y$ satisfying $p=x^2-xy+y^2$. The textbook says because when $p\equiv 2\pmod 3$ is an odd prime, $\left( \frac{-3}{p} \right) ...
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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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Eisenstein integers; finding a condition to be a unit using a multiplicative function.

Context $1$: Let $E = \Bbb Z [w] = \{m + nw \in \Bbb C \hspace{.1cm}| \hspace{.1cm} m,n \in \Bbb Z\}$ the domain of the Eisenstein integers. Doing some work, we know that $\omega^2 = \bar{\omega} $. ...
xyz's user avatar
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Factor in Eisenstein Integers.

I wonder if there is a way to factor a number in $\mathbb{Z[\omega]}$ more "general"?, I currently use $(a+b\omega)(a+b\omega^2) = a^2 - ab + b^2 = \frac{(2a-b)^2 + 3b^2}{4}$ to find for ...
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Fermat's Last Theorem ($n=3$) using the Eisenstein integers

I'm doing the first 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'm assuming I should ...
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Quotient ring $\mathbb{Z}[u]/(a+bu)$, when $a$ and $b$ are relatively prime.

It's known that when $a$ and $b$ are comprime $\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$, and $\mathbb{Z}[\omega]/(a+b\omega) \cong \mathbb{Z}_{a^2+b^2-ab}$, where $\omega$ is a primitive ...
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What does a "torus obtained by identifying each of the three pairs of opposite edges of a regular hexagon" look like? What is it called? [duplicate]

Wikipedia's Eisenstein integer; Quotient of C by the Eisenstein integers says: The quotient of the complex plane C by the lattice containing all Eisenstein integers is a complex torus of real ...
uhoh's user avatar
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Extending the Supplement of Eisenstein Reciprocity

One of the supplements of Eisenstein Reciprocity states the following: Supplement: If $m$ is an odd prime and $a$ is a rational integer relatively prime to $m$, then $\left(\frac{1-\zeta_m}{a }\right)...
Sohail Farhangi's user avatar
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Are all points on the shore of Gosper Island either transcendental or Eisenstein rational?

I'm working on a program which calculates in the flowsnake base (2.5-√-0.75, with cyclotomic digits) and I've come up with some observations and questions about this base representation. The program ...
Pierre Abbat's user avatar
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On primality of an element in the ring $\mathbb Z[\omega]$ [duplicate]

The problem is: Show that if $p(\neq 3)$ is a prime in $\mathbb Z$ and $p \neq a^2+b^2-ab$ for any $a,b \in \mathbb Z$, then $p$ is a prime element in the ring $R = \mathbb Z[\omega]$. My approach: I ...
Rabi Kumar Chakraborty's user avatar
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using eisenstein integers to solve diophantine equations

Earlier, I asked a question; this is a question regarding an answer I received to it. Apparently when $u^2 - 12p^2 = -3$, with $p$ a three digit prime, there is some sort of recursion of the values of ...
justadumbguy's user avatar
3 votes
2 answers
329 views

$x^3 + y^3 = p^2$ over the integers

$x^3 + y^3 = p^2$ has a solution over the integers for some three digit prime p. Find all p that satisfy. The first thing I did was factorize the left hand side, getting $(x+y)(x^2 - xy + y^2) = p^2$ ...
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How to prove that $\mathbb{Z}[\exp(\frac{2\pi i}{3})]$ is a Euclidean domain? [duplicate]

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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Fermat's Little Theorem for Eisenstein primes

Prove that if $\alpha \in \mathbb{E}$ is an Eisenstein integer and $\pi$ is an Eisenstein prime, than $\pi \mid \alpha^{N(\pi)}-\alpha$. $\mathbb{E} = \mathbb{Z}[\varepsilon] = \{ a+\varepsilon b \...
sicmath's user avatar
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12 votes
2 answers
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is 1001 the only sum of two positive cubes that is the product of three consecutive odd primes?

That is $\ 10^3+1^3=7.11.13$. I could find no other examples. So I am looking to see if there are any more solutions to $ x^3+y^3=p.q.r$, where $ x, y$ are positive integers and $ p<q<r$ are ...
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Exercise 0.17 in Miles Reid's Commutative Algebra

Use the result of Ex. 0.16 to deduce another proof of the fact that any prime $p\equiv_6 1$ is of the form $3a^2+b^2$. Exercise 0.16: Prove that the ring $\Bbb Z[\omega]$ is a UFD, where $\omega^2+\...
cansomeonehelpmeout's user avatar
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2 answers
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How to find all solutions for : $a^3 \equiv b^3 \pmod{7^3}$, knowing that $7 \nmid ab$.

Find all integers $a$ and $b$ such that $$a^3 \equiv b^3 \pmod{7^3}\,,$$ knowing that $7 \nmid ab$. As a try, I noticed that, since $\gcd(b, 7)=1$, there exists $x \in \mathbb{N}$ such that $b\cdot x ...
PortoKranto's user avatar
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Smallest residue over $\Bbb Z[\omega]$

I'm asked to prove that $\Bbb Z[\omega]$, where $\omega^2+\omega+1=0$, is a Euclidean domain. The norm is $N(a+b\omega)=(a+b\omega)(a+b\omega^2)$. My strategy is to write $\alpha=\beta\gamma+\rho$, ...
cansomeonehelpmeout's user avatar
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Eisenstein's Criterion with an example

Wolfram Alpha says $x^5 -x^2 +1$ is irreducible over $\mathbb{Z}$. Is there any way to prove it by Eisenstein's Criterion? I tried to translate this function. I translated the function a couple of ...
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How to show that $a^2+ab+b^2<p$ for integers $a,b$?

So I am trying to prove that $p\equiv 1\pmod{3}$ implies that there exists integers $a,b$ such that $p=a^2+ab+b^2.$ First using quadratic reciprocity we have the existence of integer $d$ such that $d^...
Student's user avatar
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7 votes
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Conductor of $\mathbb Q(\omega,\sqrt[3]{\pi})/\mathbb Q(\omega)$ for nonprimary $\pi$

I have recently been playing around with abelian extensions and I have have found myself playing with Magma and computing conductors of cubic extensions of $F=\mathbb Q(\omega)$, where $\omega$ is a ...
Wojowu's user avatar
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Prove $\mathbb{Z}[\omega]/\left\langle p\right \rangle \cong \mathbb{F}_p[x]/\left \langle x^2+x+1 \right \rangle$

Let $\omega=e^{2\pi\over 3}$ and let $R=\mathbb{Z}[\omega]$. Let $p>3$ be a prime number. Prove $\mathbb{Z}[\omega]/\left\langle p\right \rangle$ is isomorphic to $\mathbb{F}_p[x]/\left \langle x^2+...
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Associated elements in $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$

I got four elements and want to check whether these elements are associated or not. For $\alpha = \frac{1+\sqrt{-3}}{2}$ my elements are: $a_1 = 2 - \alpha = \frac{3 - \sqrt{-3}}{2}$ $a_2 = 1 - 2\...
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What are the positive integer solutions to $x^2-x+1 = y^3$?

The only solutions that I know of till now are $(x,y) = (1,1) \space , (19,7)$. We can note that: $$x^2-x+1 = y^3 \implies (2x-1)^2 = 4y^3-3$$ Thus, if odd prime $p \mid y$, then $(2x-1)^2 \equiv -3 \...
Haran's user avatar
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1 answer
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How many Eisenstein integers modulo 3 are there?

I'm trying to find a system of representatives for the Eisenstein integers modulo 3. What would the set S be and how can I determine the number of solutions in S of the equation x^2=0 mod 3?
Jingting931015's user avatar
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1 answer
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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 ...
Jingting931015's user avatar
9 votes
3 answers
737 views

The set of integers $n$ expressible as $n=x^2+xy+y^2$

Let $S$ be the set of integers $n$, such there exist integers $x,y$ with $$n=x^2+xy+y^2$$ Is the implication $$a,b\in S\implies ab\in S$$ true? If yes, how can I prove it? I worked out $$n\in S\iff ...
Peter's user avatar
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2 votes
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The idempotent elements of Eisenstein Integers

Let a+bω be an Eisenstein integer. An idempotent element of $ \mathbb Z_n[\omega]$ is $(a+b\omega)^2 \equiv (a+b\omega)\pmod{n} $, where $\omega^2=-\omega-1$ But it follows that the idempotent element ...
casey garcia's user avatar
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1 answer
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Why is the ring of Eisenstein integers interesting [closed]

Also there's quite little information on its history. Can anyone please enlighten me in this ring. Particularly i had been researching about it but I feel its not quite enough to be a motivation to be ...
casey garcia's user avatar
16 votes
3 answers
2k views

Factoring rational primes over the Eisenstein integers - when can a prime be written as $j^2+3k^2$?

I've been messing around with Eisenstein integers, and comparing them with Gaussian integers. Many things are clear, but I'm struggling with the details underlying which rational primes split, and ...
G Tony Jacobs's user avatar
1 vote
1 answer
260 views

Units of $\mathbb{Z}[\omega]$ where $\omega = \frac{1}{2} (1+ \sqrt{a})$, $a<0$ a square free integer.

Wondering about this question, which is a sort of generalized form of the Eisenstein integers, let: $$ \omega = \frac{\sqrt{a}+1}{2} $$ Where $a \equiv 1 \textrm{ mod 4}$ and $a<0$. How does one ...
rednexela1941's user avatar
2 votes
0 answers
2k views

On proving that $\mathbb{Z}[\omega]$ is a Euclidean domain.

To connclude the proof of an Eisenstein integer to be an Euclidean domain I need to show that $N(r)<N(\beta)$ where I have this assumption $|t-p|\leq\frac{1}{2}$ and $|s-q|\leq\frac{1}{2}$, $r=\...
MindSweeper's user avatar
5 votes
1 answer
536 views

Natural generalizations of Gaussian & Eisenstein integers?

$\newcommand{\iu}{{i\mkern1mu}}$Gaussian integers are complex numbers $a + b \iu$ where $a$ and $b$ are integers, and $\iu^2 = -1$. Eisenstein integers are complex numbers $a + b \omega$ where $\...
Joseph O'Rourke's user avatar
3 votes
1 answer
212 views

Eisenstein Integers

Tag description says the tag is for questions about the Eisenstein Integers. Apologies for the question. I'd like to be a bit more informed about what they are related to, and what is the motivation....
user76568's user avatar
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9 votes
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Are there further gaps in the Eisenstein primes?

I recently played around with Eisenstein primes a bit (in an admittedly very amateurish way) and noticed among other things that there are no primes on the hexagonal ring that goes through (8,0) on ...
Martin Ender's user avatar
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16 votes
6 answers
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Primes congruent to 1 mod 6

I came across a claim that I found interesting, but can't seem to prove for some reason. I have the feeling it should be easy a prime $p$ can be written in the form $p = a^2 -ab +b^2$ for some $a,b\...
Math2012pc's user avatar
5 votes
3 answers
627 views

Understanding a congruence relation in $\mathbb{Z}[\omega]$

I'm having some difficulty understanding the relation between two different congruences I've been dealing with. These come from Exercise 25 of Chapter 3 in Ireland and Rosen's Number Theory. Let $\...
yunone's user avatar
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