Questions tagged [eisenstein-integers]

For questions about Eisenstein integers.

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7
votes
0answers
34 views

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 ...
0
votes
2answers
47 views

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+...
3
votes
1answer
39 views

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\...
6
votes
1answer
199 views

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 \...
1
vote
1answer
88 views

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?
1
vote
1answer
31 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 ...
1
vote
1answer
43 views

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 ...
0
votes
1answer
58 views

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 ...
14
votes
3answers
386 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 ...
0
votes
1answer
63 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 ...
0
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0answers
421 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=\...
3
votes
1answer
184 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 $\...
2
votes
1answer
84 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....
7
votes
1answer
169 views

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 ...