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

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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 \...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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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### 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 ...
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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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### 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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### 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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### 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 ...
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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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### 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 ...
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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"...
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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 ...
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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 ...
### 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 ...
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 ...