Questions tagged [quadratic-integer-rings]

Use this tag for questions related to the subset of quadratic integers contained in a quadratic field.

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23 views

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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0answers
32 views

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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0answers
30 views

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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1answer
43 views

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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1answer
174 views

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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1answer
43 views

Minimal generating set of an ideal in a polynomial ring

Suppose $R$ is a ring and $I = \langle a, b \rangle$ is a non-principal ideal of $R$. Does the ideal $J = \langle x, a, b \rangle$ of the polynomial ring $R[x]$ require three generators? How about ...
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2answers
78 views

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? ...
2
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1answer
59 views

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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1answer
78 views

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 ...
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0answers
77 views

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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2answers
112 views

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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1answer
36 views

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 ...
2
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1answer
69 views

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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3answers
87 views

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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0answers
12 views

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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1answer
160 views

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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1answer
201 views

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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4answers
100 views

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 ...
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1answer
104 views

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[\...
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1answer
118 views

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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2answers
241 views

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 ...
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3answers
2k views

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 ...
3
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1answer
510 views

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 ...
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1answer
534 views

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}...
3
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0answers
108 views

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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1answer
84 views

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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2answers
445 views

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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1answer
47 views

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 ...
42
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6answers
5k views

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=2,3\ \pmod 4\\ \mathbb{Z}\left[\frac{1+\sqrt{D}}{2}\...
5
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0answers
225 views

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 ...
1
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1answer
416 views

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"...
3
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1answer
135 views

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 ...
5
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3answers
246 views

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 ...
2
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1answer
245 views

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 ...
6
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3answers
1k views

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 ...