Questions tagged [quadratic-integer-rings]
Use this tag for questions related to the subset of quadratic integers contained in a quadratic field.
47
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Prove that $\mathcal{O}$ is a Euclidean Domain.
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 $\...
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0
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Irreducible density race : ring $A = \Bbb Z[\frac{1+\sqrt {-23}}{2}]$ vs ring $B = \Bbb Z[\frac{1+\sqrt {-31}}{2}]$?
Inspired by this question here :
Prime density race : ring $A = \Bbb Z[\frac{1+\sqrt {-7}}{2}]$ vs ring $B = \Bbb Z[\frac{1+\sqrt {-11}}{2}]$?
I now consider the smallest cases of non-UFD.
I did not ...
- 15.3k
0
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0
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The number of imaginary quadratic fields of class number 3 is finite ??
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.
...
- 15.3k
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1
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Prime density race : ring $A = \Bbb Z[\frac{1+\sqrt {-7}}{2}]$ vs ring $B = \Bbb Z[\frac{1+\sqrt {-11}}{2}]$?
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 ...
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0
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The bound $\prod\limits_{i=0}^ju_i<\alpha_j<\prod\limits_{i=0}^j(u_i+1)$ not correct?
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 ...
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1
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Abelian subgroup $(M,+) \subset ( \mathbb{Z}[\alpha],+)$ is of the form $m\mathbb{Z}\oplus(a+b\alpha) \mathbb{Z}$
$\mathbb{Z}[\alpha]$ 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 ...
1
vote
1
answer
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How to show the quadratic integer ring O is not a UFD.
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 < ...
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1
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Description about the set of quadratic integer with prime norm. [closed]
I have a guess that, if $p$ is a prime number, then
$$ \text{if }\exists a\in\mathbb{Z}[\sqrt{k}] \text{ such that } N(a)=p,
\\ \text{ then }\left\{z\in \mathbb{Z}[\sqrt{k}]:N(z)=p\right\}= \left\{ a,...
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1
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0
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Conjugation map in quadratic integer rings is an automorphism
Let $d$ be a square free integer such that $d \equiv 1 \mod 4$ and let $R$ be the ring
$$
R = \left\{a + b\frac{1+\sqrt{d}}{2} \mid a, b \in \mathbb{Z}\right\}
$$
I am trying to show that the map
\...
- 575
1
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0
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Understanding the difference between 2 questions.
Here is the question I want to answer:
Let $R$ be the quadratic integer ring $\mathbb Z[\sqrt{-5}]$ and define the ideals $I_2 = (2, 1 + \sqrt{-5}), I_3 = (3, 2 + \sqrt{-5}),$ and $I_3^{'} = (3, 2 - \...
- 893
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1
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Factorizations in $\mathbb{Z}[\theta]$
I'm trying to comprehend a proof from my Elementary Number Theory course. Here, a quadratic integer $\theta$ is a solution of an equation of the form $x^2 + bx + c = 0$ with $b$ and $c$ integers.
Let ...
- 1,525
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4
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Finding the units of $\mathbb{Z}\left[\frac{1+\sqrt{-19}}{2}\right]$
I was trying to comprehend a simple exercise from my elementary number theory class.
Let $\theta=\frac{1+\sqrt{-19}}{2}$, and let $R$ be the ring $\mathbb{Z}[\theta] .$ Show that the units of $R$ are ...
- 1,525
2
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1
answer
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Solving $n(4n+3)=2^m-1$ in positive integers
Find all positive integers $m$ and $n$ such that $$n(4n+3)=2^m-1\,.$$
This is an interesting equation which was sent to me by a friend (probably found online). I have been scratching my head about ...
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0
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Quadratic Integer Ring mod p - Field?
In order to understand a proof of a book I try to get I need to deal with Quadratic Integer Rings. As far as I got till now if I look at $\mathbb{Q}(\sqrt(d))$, $O_{\sqrt(d)}$ and $p \equiv \eta_p \...
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Quadratic Integer Ring $A=\mathbb{Z}[\sqrt{-14}]$, show that $7$ is irreducible in A.
I will proceed with a proof by contradiction.
If we assume that $7$ is reducible, then there exists some $q$ and $r$ in $A$ such that $q$ and $r$ are not units, and $7 = qr$. Using the norm ( $N(a+b\...
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Quadratic Integer Ring $\mathbb{Z}[\sqrt{-14}]$, show that $7+\sqrt{-14} \notin (7)$
I just want to make sure I am getting this correct, as I am just learning ring theory and quadratic integer rings.
Let $\alpha = 7 + \sqrt{-14}$.
To show that $\alpha \notin (7)$ I will show a ...
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1
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Prime and Maximal Ideals of $\mathbb{Z}[\sqrt{-10}]$
I want to solve the following questions:
For $R = \mathbb{Z}[\sqrt{-10}]$ and $I = (\sqrt{-10})$
I ) Is $I$ a prime ideal?
II) Is $I$ a maximal ideal?
This is equivalent to asking if $\frac{R}{I}$ ...
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1
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Show $\mathbb{Z}_{(p)} [ \sqrt{D}]$ is a UFD
Consider $\mathbb{Z}_{(p)} [ \sqrt{D}]$, for $D$ a squarefree integer, and $D \not\equiv 1 \bmod 4$. I want to show that this is a UFD.
By considering $\mathbb{Z}_{(p)} [ \sqrt{D}] \cong (\mathbb{Z} ...
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2
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For prime quadratic integer $\pi$, $x \equiv 1$ (mod $\pi$), Show $x^2 \equiv1$ (mod $\pi^2$) and $x^3 \equiv 1$ (mod $\pi^3$) is not always true.
I was working through one of the problems given to me on my problem set for a number theory class and I would like some help in an attempt to learn. Could someone help me with the following question?
...
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2
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1
answer
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Counting number of ideals in quadratic number field
Let $K$ be a quadratic number field and $R$ be its number ring, and if $a(n)$ denotes number of ideals of norm $n$, if $n$ is a prime number, then number of ideals of norm $n$ is $1+(d|n)$, where $d$ ...
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1
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How to approximate real numbers using members of $\mathbb Z (\sqrt d)$?
Real numbers can be approximated to successively better precision using the convergents of a continued fraction.
Is there a similar way to find quadratic integers of fixed (positive) discriminant ...
- 53
5
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0
answers
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How to interpret action of $SL_2(\mathcal{O}_d)$
Given a lattice $\wedge = \{\omega_1, \omega_2 \}$ in $\mathbb{C}$, $\omega_1 / \omega_2 \not\in \mathbb{R}$, we know that $\wedge' = \{\omega_1', \omega_2' \}$ defines the same lattice precisely when
...
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2
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Recognizing Principal Ideals
In the ring $\mathbb{Z}[\sqrt{6}]$, the ideal $(2,\sqrt{6})$ simplifies to $(\sqrt{6}-2)$, while in the ring $\mathbb{Z}[\sqrt{10}]$, the ideal $(2,\sqrt{10})$ is not principal (I think). Is there ...
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Factoring an integer over quadratic ring
The discriminant of the ring $\mathbb{Z}[\sqrt{10}]$ is 40. Since 2 divides 40, 2 is ramified in this ring. I know that the ideal $(2)$ factors as $(2,\sqrt{10})^2$, but I cannot find a factorization ...
- 978
2
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1
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Relatively Prime Integers Still Prime in Quadratic Fields
The problem I was given: "If $m$ and $n$ are relatively prime rational integers, must they be relatively prime in every quadratic field $\mathbb{Q}[\sqrt d]$? If so, give a justification, if not, give ...
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Prove that $\langle 2,1+\sqrt{-5} \rangle ^ 2 \subseteq \langle 2 \rangle$
I've got an exercise whose goal I pressume is to show how $\mathbb{Z}[\sqrt{-5}]$ is not a PID. They define $I = \langle 2,1+\sqrt{-5} \rangle$ then they show that $I$ is a maximal ideal on $\mathbb{...
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Some reference on quadratic rings or introductory algebraic number theory. [duplicate]
I have found many posts on this matter. But none of them fulfills my purpose. So I am describing my situation briely. I have read ring theory from Dummit & Foote. I have been reading some field ...
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Prove that the ideal $I = \left( 3, 2 + \sqrt{-5} \right)$ is a prime ideal in $\mathbb{Z}\left[ \sqrt{-5} \right]$.
Prove that the ideal $I = \left( 3, 2 + \sqrt{-5} \right)$ is a prime ideal in $R = \mathbb{Z}\left[ \sqrt{-5} \right]$.
The book recommends observing that $$ R/I \cong \left( R/(3) \right)/\left( I/...
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1
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Ideals of the quadratic integer ring $\mathbb{Z}[\sqrt{-5}]$
Can someone please explain the contradiction when we take $a^2 + 5b^2$ to be equal to $1$.
After multiplying both sides by $2 - \sqrt{-5}$ and factoring at a $3$ I get $$3\left[\left(2-\sqrt{-5} \...
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4
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Existence of integers $a$ and $b$ such that $p = a^2 +ab+b^2$ for $p = 3 $ or $p\equiv 1 \mod 3$
**Eisenstein primes, existence of integers **
I am working in subring $$R = \{a + b\zeta : a,b \in \mathbb{Z}\}$$ of $\mathbb{C}$ where $\zeta = \frac{1 + \sqrt{-3}}{2} \in \mathbb{C}$. I want to ...
- 1,073
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1
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Quotient of quadratic integer by a maximal ideal
I find the following proposition in my class notes (ring theory) without proof, not sure if I have wrote it down correctly .
Let $P$ be a maximal ideal in the ring of integers $Z[\sqrt{d}]$, then $Z[\...
0
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1
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Quadratic Integers and Sandwich Numbers [closed]
Let's define a sandwich number such that it is sandwiched between a perfect square and a perfect cube. For example, 26. $26-1=5^2$ and $26+1=3^3$. We can use the equation $y^3-x^2=2$ to define these ...
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Why is the fact that $n$ is square free important to consider?
Let $N_{1}$ be the set of all positive integer greater than $1$.Let $n \in N_{1}$ is square free. Let us consider $\mathbb Z [\sqrt{n}] = \{a + b {\sqrt{n}} : a,b \in \mathbb Z \}$, for some square ...
user251057
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Fundamental unit of the quadratic integer ring $\mathbb{Z}[\sqrt{n}]$
I looked in other questions but I didn't find any answers regarding quadratic integer rings. Apologies if I missed it.
Given the ring $\mathbb{Z}[\sqrt{n}]$ where $n$ is a square-free positive ...
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1
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How do I prove that $\mathbb{Z}[\sqrt{19}]$ is not a UFD?
How do I prove that $\mathbb{Z}[\sqrt{19}]$ is not a UFD? Moreover, how do I prove that $(7,3+\sqrt{19})$ is not a principal ideal?
This is the first time I'm dealing with a quadratic integer ring ...
- 12.4k
1
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1
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Prove a property of the order of conductor $f$ in the field $\mathbb{Q}(\sqrt{D})$
Let $D$ be a squarefree integer, and let $\mathcal{O}$ be the ring of integers in the quadratic field $\mathbb{Q}(\sqrt{D})$. For positive integer $f$ define the order of conductor $f$, $\mathcal{O}...
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0
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Lists of negative discriminants by class group?
Is there a handy listing of the discriminants of imaginary quadratic fields having a given ideal class group? It would be nice to use such a resource as a source of examples.
For example, we're all ...
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1
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Show that $ζ$ is a Quadratic Integer in $Q[\sqrt{−3}]$
So in the complex plane, there are three cube roots of one. Suppose we let $ζ$ be the cube root of one which has positive imaginary part. How can we show that $ζ$ is a quadratic integer in $Q[\sqrt{−3}...
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Enumerating Bianchi circles
Background: Katherine Stange describes Schmidt arrangements in "Visualising the arithmetic of imaginary quadratic fields", arXiv:1410.0417. Given an imaginary quadratic field $K$, we study the Bianchi ...
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1
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How do I prove that $Z[w]/(a)\cong Z_{N(a)}$?
Define $S=\{\sqrt{D}:D\equiv 2,3 \pmod 4\}$
Define $T=\{\frac{1-\sqrt{D}}{2}:D\equiv 1 \pmod 4 \}$.
Let $w\in S\cup T$.
Let $N:\mathbb{Z}[w] \rightarrow \mathbb{Z}$ be the norm.
Then how do I ...
- 12.4k
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6
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Why is quadratic integer ring defined in that way?
Quadratic integer ring $\mathcal{O}$ is defined by
\begin{equation}
\mathcal{O}=\begin{cases}
\mathbb{Z}[\sqrt{D}] & \text{if}\ D\equiv2,3\ \pmod 4\\
\mathbb{Z}\left[\frac{1+\sqrt{D}...
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0
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What is an importance of Gaussian and Eisenstein rings?
Among quadratic integer rings, $\mathbb{Z}[i]$ and $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$ have their special names, namely Gaussian integers and Eisenstein integers respectively. I guess this is named so ...
- 12.4k
1
vote
1
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Prime element of a quadratic integer ring
For simplicity, let $D$ be a square-free integer such that $D\equiv 2,3 \pmod 4$ such that the class number of $\mathbb{Z}[\sqrt{D}]$ is 1.
-Theorem in the page for "quadratic integer rings"...
- 12.4k
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1
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How do I determine ring types of quadratic integers -43,-67,-163?
Let $D=-43,-67,-163$.
Then $\mathbb{Z}[\frac{1+\sqrt{D}}{2}]$ is not a Euclidean domain since it does not have a universal side divisor, but how do I prove that this is a PID?
Moreover, I have shown ...
- 12.4k
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3
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Good reference book for quadratic integer rings?
Could anyone direct me to a good reference book(s) for quadratic integer rings? Ideally, the reference would begin with their elementary properties and then proceed through their ring-theoretic ...
user123641
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1
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Prove $\Bbb{Z}[\sqrt{d}] /(a + \sqrt{d}) \cong\Bbb{Z}/n\Bbb{Z}$, where $n=|a^2-d|$.
Let $a,d$ be integers with $d$ square free. Prove that $\mathbb{Z}[\sqrt{d}]/(a + \sqrt{d}) \cong \mathbb{Z}/n\mathbb{Z}$ where $n= |a^2- d|.$
I've tried attempting the problem by looking for a ...
- 1,109
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votes
3
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Infinitely many primes in the ring of integers for any quadratic field
Let $d$ be an integer, not a perfect square, and $\mathcal{O}_K$ the ring of integers in $K = \mathbb Q(\sqrt d)$. I want to prove that there are infinitely many primes in $\mathcal{O}_K$.
How do we ...
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