Questions tagged [quadratic-integer-rings]

Question 1

The Idoneal numbers have the property that if $d$ is an Idoneal number, then there is a congruence relation on primes $p$ such that $p$ satisfying the congruence relation iff $p = x^2 + dy^2$ has a ...

Question 2

Let $\beta$ be a complex number satisfying $\beta^{2}=n_{0}+n_{1}\cdot\beta$ with integer $n_{0}$, $n_{1}$. When there exists a number of the form $w=w_{0}+w_{1}\cdot\beta$ with integer $w_{0}$, $w_{1}...

Question 3

Consider the Euclidean complex quadratic rings. There are only a $5$ of them. $$ \Bbb Z[\sqrt{-1}],\Bbb Z[\sqrt {-2}], \Bbb Z[\frac{1+\sqrt {-3}}{2}], \Bbb Z[\frac{1+\sqrt {-7}}{2}], \Bbb Z[\frac{1+\...

Question 4

The angle in a regular $n$-dimensional simplex is $\theta_n=\arccos(1/n)$. Is there a (finite) list of integers $c_2,c_3,c_4,\cdots$ such that $$c_2\theta_2+c_3\theta_3+c_4\theta_4+\cdots=0,$$ other ...

Question 5

We know that the norm of an algebraic integer is an integer, so I want to go a step further and know how many quadratic algebraic integers there are with a specific norm. Of course, I want to ...

Question 6

Here is the question I am trying to solve (Dummit & Foote, 3rd edition, Chapter 8, section 1, #8(a)): Let $F = \mathbb Q(\sqrt{D})$ be a quadratic field with associated quadratic integer ring $\...

Question 7

I read here The primes $p$ of the form $p = -(4a^3 + 27b^2)$ that " It is known that the number of imaginary quadratic fields of class number 3 is finite. " But the links did not show it. ...

Question 8

Consider the ring $A = \Bbb Z[\frac{1+\sqrt {-7}}{2}]$ with the elements $\frac{a+b\sqrt {-7}}{2}$ where $a,b$ are both even or both odd integers. This is also known as the Kleinian integers; the ring ...

Question 9

In Distribution of class numbers in continued fraction families of real quadratic fields Lemma 8 says let $\omega_d=[u_0,\overline{u_1,u_2,\dots ,u_{s-1},u_s}]$ (note: only re-specified in the ...

Question 10

$\mathbb{Z}[\alpha]$ is the quadratic integer ring associated to the squarefree integer $d$. Let $m\in M \cap\mathbb{Z}$ and $\beta=a+b\alpha\in M$. If $\delta=x+y\alpha$, write $y=qb+r$, where $0\le ...

Question 11

Let $R=\mathbb{Z}[\sqrt{−n}]$ where $n$ is a squarefree integer greater than 3. Prove that $R$ is not a UFD. Conclude that the quadratic integer ring O is not a UFD for $D\equiv 2, 3$ mod $4$, $D < ...

Question 12

I have a guess that, if $p$ is a prime number, then $$ \text{if }\exists a\in\mathbb{Z}[\sqrt{k}] \text{ such that } N(a)=p, \\ \text{ then }\left\{z\in \mathbb{Z}[\sqrt{k}]:N(z)=p\right\}= \left\{ a,...

Question 13

Let $d$ be a square free integer such that $d \equiv 1 \mod 4$ and let $R$ be the ring $$ R = \left\{a + b\frac{1+\sqrt{d}}{2} \mid a, b \in \mathbb{Z}\right\} $$ I am trying to show that the map \...

Question 14

Here is the question I want to answer: Let $R$ be the quadratic integer ring $\mathbb Z[\sqrt{-5}]$ and define the ideals $I_2 = (2, 1 + \sqrt{-5}), I_3 = (3, 2 + \sqrt{-5}),$ and $I_3^{'} = (3, 2 - \...

Question 15

I'm trying to comprehend a proof from my Elementary Number Theory course. Here, a quadratic integer $\theta$ is a solution of an equation of the form $x^2 + bx + c = 0$ with $b$ and $c$ integers. Let ...

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Questions tagged [quadratic-integer-rings]

What is the largest integer $d$ such there is a congruence relation on primes p so that $x^2 + dy^2 = p^2$ has a non-zero integral solution?

Let $\beta^{2}=n_{0}+n_{1}\cdot\beta$ and $w=w_{0}+w_{1}\cdot\beta$; when $\sqrt{w}$, $\sqrt{w+1}$, $\sqrt{w+2}$ have the same form?

Patterns for Euclidean complex quadratic rings?

Is there any relation between the dihedral angles in the regular triangle (60 degrees), tetrahedron (70.53), pentachoron (75.52), etc.?

The number of non-real quadratic algebraic integers with the given norm in the first quadrant and the upper z-axis.

Prove that $\mathcal{O}$ is a Euclidean Domain.

The number of imaginary quadratic fields of class number 3 is finite ??

Prime density race : ring $A = \Bbb Z[\frac{1+\sqrt {-7}}{2}]$ vs ring $B = \Bbb Z[\frac{1+\sqrt {-11}}{2}]$?

The bound $\prod\limits_{i=0}^ju_i<\alpha_j<\prod\limits_{i=0}^j(u_i+1)$ not correct?

Abelian subgroup $(M,+) \subset ( \mathbb{Z}[\alpha],+)$ is of the form $m\mathbb{Z}\oplus(a+b\alpha) \mathbb{Z}$

How to show the quadratic integer ring O is not a UFD.

Description about the set of quadratic integer with prime norm. [closed]

Conjugation map in quadratic integer rings is an automorphism

Understanding the difference between 2 questions.

Factorizations in $\mathbb{Z}[\theta]$

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