Questions tagged [eisenstein-integers]

For questions about Eisenstein integers.

Eisenstein integers or, Eisenstein-Jacobi integers or, Eulerian integers are defined to be the set $$~\mathbb Z[\omega] = \{a + b\omega : a, b \in \mathbb Z\}~$$ where $$~\omega = \frac{1}{2}(−1 + i \sqrt 3)=e^{2\pi i/3}~.$$

This set lies inside the set of complex numbers $~\mathbb C~$ and they form a commutative ring in the algebraic number field $~\mathbb Q(\omega)~$.

Note:

$1.~$ Like the complex plane is partitioned symmetrically into four quadrants, the Eisenstein integers is symmetrically and radially partitioned into six sextants. Each sextant is defined as follows.

  • First sextant: $~ \left\{\eta \in \mathbb Z[\omega] \mid 0 \le \operatorname{Arg}(\eta) <\frac{\pi}{3}\right\} ~$
  • Second sextant: $~ \left\{\eta \in \mathbb Z[\omega] \mid \frac\pi3 \le \operatorname{Arg}(\eta) <\frac{2\pi}{3}\right\} ~$
  • Third sextant: $~ \left\{\eta \in \mathbb Z[\omega] \mid \frac{2\pi}3 \le \operatorname{Arg}(\eta) <\pi\right\}~$
  • Fourth sextant: $~\left\{\eta \in \mathbb Z[\omega] \mid -\pi < \operatorname{Arg}(\eta) < -\frac{2\pi}3 ~~ \text{or,$~~$}\operatorname{Arg}(\eta) = \pi\right\}~$
  • Fifth sextant: $~\left\{\eta \in \mathbb Z[\omega] \mid -\frac{2\pi}3 \le \operatorname{Arg}(\eta) <-\frac{\pi}{3}\right\}~$
  • Sixth sextant: $~\left\{\eta \in \mathbb Z[\omega] \mid -\frac\pi3 \le \operatorname{Arg}(\eta) <0\right\}~$

$2.~$ Eisenstein integers form a unique factorization domain.

More information can be found in this Wikipedia article.

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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 $a,b\in\mathbb{Z}$ if and only if $p\equiv 1\bmod{6}$
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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 which are inert. I know that the inert primes are…
G Tony Jacobs
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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
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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 prove it using the Eisenstein integers $\Bbb…
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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 $(2x-1)^2 \equiv -3 \pmod{p}$ and thus, $-3$ is a quadratic residue.…
9
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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 $$n\in S\iff 4n\in S$$ and $$n\in S\iff 3n\in S$$ I tried…
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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 the Eisenstein grid of the complex plane: I…
Martin Ender
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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 primitive third root of unity. I have stumbled…
Wojowu
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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 $\omega= e^{2 \pi i/3}$, so that $\omega^3 = 1$, i.e.,…
Joseph O'Rourke
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$x^3 + y^3 = p^2$ 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 $(x+y)(x^2 - xy + y^2) = p^2$ I then considered the case $x^2 - xy + y^2 =…
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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 $\pi \mid \alpha^{N(\pi)}-\alpha$. $\mathbb{E} = \mathbb{Z}[\varepsilon] = \{ a+\varepsilon b \mid a, b \in \mathbb{Z} \}$ is the ring of the…
sicmath
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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: $a_1 = 2 - \alpha = \frac{3 - \sqrt{-3}}{2}$ $a_2 = 1 - 2\alpha = -\sqrt{-3}$ $ a_3 = 3 + 2\alpha = 4 +…
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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. Trying to single out what and why my self mostly…
user76568
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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 $R = \mathbb{Z}[\rho]$ together with $N(z) = |z|^2$, for $z \in R$, is…
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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 \equiv 1 \pmod{7} \Rightarrow b^3 \cdot x^3…
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