Questions tagged [euclidean-domain]

Use for questions related to commutative rings that can be endowed with a Euclidean function, which allows a suitable generalization of the Euclidean division of the integers.

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61 views

Smallest residue over $\Bbb Z[\omega]$

I'm asked to prove that $\Bbb Z[\omega]$, where $\omega^2+\omega+1=0$, is a Euclidean domain. The norm is $N(a+b\omega)=(a+b\omega)(a+b\omega^2)$. My strategy is to write $\alpha=\beta\gamma+\rho$, ...
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0answers
38 views

Prove that $\mathcal{O}_K$ for $K = \mathbb{Q}(\sqrt{3})$ is euclidean

I'm having trouble with this question concerning the ring of integers of a real quadratic field: I know that if we can prove that the ring is euclidean then every ideal is principal, and thus $C(\...
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1answer
58 views

When is $\mathbb Z[\sqrt d]$ a Euclidean or principal ideal domain?

Let $d$ be an integer $\neq 1$ such that $d$ is not divisible by the power of any prime.Consider the ring $\mathbb Z[\sqrt d]$.My question is when is this ring a Euclidean domain,and when it is not an ...
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2answers
22 views

Find the equation of the two tangent planes to the sphere $x^2+y^2+z^2-2y-6z+5=0$ which are parallel to the plane $2x+2y-z=0$

Find the equation of the two tangent planes to the sphere $x^2+y^2+z^2-2y-6z+5=0$ which are parallel to the plane $2x+2y-z=0$ My Attempt We need to find a point which is shortest distance from the ...
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41 views

Proving Gaussian Integers as an Euclidean Domain

I was going through this link for the proof "Gaussian Integers being an Euclidean Domain" with the function $f(z)=|z|^2$ https://www.cmi.ac.in/~shreejit/Gaussian.pdf We have $$Z[i]=\left\{a+...
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2answers
27 views

Divisibility in a Euclidean Domain

Let $R$ be a Euclidean Domain. I am working on showing that $$ \text{If } a \, | \, bc \text{ with } a,b \neq 0 \text{ then } \frac{a}{(a,b)} \, \bigg| \, c. $$ Note that the first part of this ...
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1answer
16 views

Minimality condition while proving ED is a PID.

In proving the following statement: Every Euclidean domain is a Principal Ideal Domain. We are trying to show that every non-zero ideal in Euclidean Domain is a Principal Ideal. So, let $A(\neq 0)$ ...
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1answer
28 views

Conditions for an euclidean domain to be a field

I am having trouble proving the following. Let $R$ an Euclidean domain with degree function $\delta,$ i.e., $\delta(ab)=\delta(a)\delta(b)$ for all $a,b\in R-\{0\}$ and $\delta(a+b)\leq\textrm{...
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1answer
26 views

How do I solve the below exercise on subset theory

Let $A\subset$ $\mathbb{R}$$^{n}$ be given. Show that there is a smallest closet set $\bar{A}$, which contains A: I.e., $\bar{A}$ is a closed set such that $A\subset$$\bar{A}$ and if C is a closed se ...
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1answer
28 views

Normalized valuations of $A$ (corresponding to the prime elements of $A$).

So, while preparing my Algebra final, i found a paper about a generalization of the idea of an euclidean ring, and thought that reading it would be enriching. However, in certain corollary (corollary ...
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16 views

Is a normalized cosine similarity a Bregman divergence?

A Bregman divergence is defined as $D(p,q) = F(p) - F(q) - < \nabla F(q), p-q>$ with F a strictly convex function of the Legendre type. Squared Euclidian distance is a Bregman divergence, with $...
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37 views

Why must a Euclidean function map to $\mathbb{Z}^{\ge 0}$?

I'm not sure I get the motivation for a Euclidean function having to map to $\mathbb{Z}^{\ge 0}$. E.g. it would seem that $\mathbb{R}^{\ge 0}$ would be a natural choice for a ring of "polynomials" ...
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4answers
68 views

How to solve $10x^{39 }+ 8x^{20} + 9x^3 + 7x ≡ 0\pmod {19}$

I guess I'm supposed to solve it with Euclidian ring (or Euclidian domain), but I'm not even sure about that. $$10x^{39 }+ 8x^{20} + 9x^3 + 7x ≡ 0\pmod {19}$$ I've managed to turn it into $(10x^{19}+...
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1answer
46 views

Prove if $R$ is not a field then the value set $δ(R)$ is infinite, $δ$ a Euclidean degree function.

I was trying to solve this exercise in my course notes, but the statement didn't seem right to me. When looking at the ring $\mathbb{Z}/4\mathbb{Z}$, it is clearly not a field since $2 + \mathbb{Z}$ ...
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30 views

For proving Euclidean domain, how to decide the mapping?

Whenever I get questions to prove it an "Euclidean domain(modern algebra)", I get confused in selecting the mapping. Is there any way to choose the mapping? Eg. If the question is to prove The ring ...
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1answer
19 views

Norm function of ED

While proving that $\langle 3,2+\sqrt{-5}\rangle$ is not a principal ideal of $\mathbb{Z}[\sqrt{-5}]$, the norm function $N(a+b\sqrt{-5})=a^2+5b^2$ has been used to arrive at a contradiction. This in ...
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2answers
71 views

Is $\frac{\mathbb{Z}[X,Y]}{(2, X)}$ a Euclidean Domain?

Is $\frac{\mathbb{Z}[X,Y]}{(2, X)}$ a Euclidean Domain? Definition for a Euclidean Domain (ED) I tend to use is the following: Let $R$ be an integral domain and $R$ supports at least one Euclidean ...
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1answer
29 views

Diophantine in Q[x]

Suppose $f, g \in Q[x]$ non-zero elements. Let $(f, g) = (d),$ and $h \in (d).$ Then there exit polynomials $p,q$ such that $h =pf + gf.$ I want to show that $h= p^{\prime}f +q^{\prime}g$ iff $p^{\...
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0answers
23 views

Lipschitz domain measurable

Is it true that any Lipschitz domain $\Omega\subset \mathbb{R}^n$ (having the boundary locally the graph of a Lipschitz function) is Lebesgue measurable?
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I need to estimate how narrow a passage is for a point surrounded by obstacles.

I find the nearest obstacle from a given point. For the obstacle on the opposite side which along the perpendicular between the given point and the obstacle in the first case, how do I find this point?...
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1answer
41 views

graphs and euclidean space

So, was reading an online blog and it said most real world data comes in the form of graphs and do not lie in a Euclidean space. I am not able to fully appreciate this comment. What are the ...
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1answer
15 views

A question about right continuous functions on $\mathbb{R}^d$.

I am reading Rick Durrett's Probability Theory and Examples. In Chapter 1 of the book (page 6), there are some contents about right continuous on $\mathbb{R}^d$. It says (ii) $F$ is right ...
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1answer
51 views

Prove that an ideal of a subring of $\mathbb{Q}$ is principal

Let $R$ be a subring of $\mathbb{Q}$ containing $1$. We denote the absolute value of an integer $n$ by $|n|$. Let $I$ denote a non-zero ideal of $R$. Let $0\ne b,a\in \mathbb{Z}$ be such that $\frac{...
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0answers
46 views

Closure and interior in subset topology

Let $\tau $ be a topology on set $\mathbb{R} \setminus \mathbb{Q}$ with subset topology from $\mathbb{R}$ (with Euclidean topology). Find clousure and interior sets: $A= \lbrace \pi * (1 + \frac{1}...
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15 views

Example of a smooth, bounded domain with $(x-x_0)\cdot \nu \leq 0$

does there exist a smooth, bounded domain $\Omega \subset \mathbb R^{n \geq 2}$ with $$(x-x_0) \cdot \nu(x) \leq 0 \quad \forall x \in \partial \Omega$$ for some point $x_0 \in \Omega$, where $\nu$ ...
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1answer
49 views

Definition of a Euclidean Domain: how to show that the condition $f(ab)\ge f(a)$ is actually not necessary? [duplicate]

Quoted from this question. We called an integral domain $R$ a Euclidean domain if there exists a function $f$ from $R$ to strictly positive integers such that: 1) For $a,b$ non zero in $R$, $f(ab)...
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2answers
75 views

Let $R$ be an integral domain and $p,q \in R$. Is it true that $(pq)R = pR \cap qR$?

Let $R$ be an integral domain, i.e., a commutative ring with 1 that has no-zero divisors other than 0. Let $p,q \in R$. under what conditions can we ensure that $(pq)R = pR \cap qR$, where $(pq)R$ is ...
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1answer
42 views

Proving rationals of the form $2^{-n}m$ are a Euclidean ring

I am trying to come up with a proof that numbers of the form $2^{-n}m$, where $~m \in \Bbb Z,~n \in \Bbb Z^+$ are a Euclidean ring/domain. The division property is easily verifiable (although the ...
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22 views

Does anyone know when/in what work Dedekind introduced the concept of a Euclidean domain?

I've found Dedekind referenced as the progenitor of the concept but I cannot find the source for the concept.
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1answer
262 views

Proof of: if A is an integral domain, then the units of A[x] are the constant polynomials that are units of A [duplicate]

I've a question: How do I proof that if A is an integral domain, then the units of A[x] are the constant polynomials that are units of A? I'm really confused here and I'd be really glad if someone ...
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2answers
333 views

Prove that $\mathbb{Z}[\sqrt{-2}]$ is a Euclidean domain and $\mathbb{Z}[\sqrt{-10}]$ is not

Prove that $\mathbb{Z}[\sqrt{-2}]$ is a Euclidean domain and $\mathbb{Z}[\sqrt{-10}]$ is not. I know that in general, to prove something is a Euclidean domain, I must prove the existence of a ...
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0answers
38 views

Why the norm function in the Euclidean domain is taken to be integer valued?

I have just started learning the Euclidean domain in Ring Theory. My book defines the norm function as follows: Let $R$ be an integral domain. Any function $N\colon R\to\mathbb{Z^{+}}\cup\{0\},$ ...
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1answer
86 views

Prove that $R=\mathbb{Z}[i]$ is a Euclidean domain via $N(a+bi) = a^2+b^2. $

Prove that $R=\mathbb{Z}[i]$ is a Euclidean domain via $N(a+bi)=a^2+b^2$ We want to show that $\forall a,b\neq 0\in\mathbb{Z}[i],\exists q,r\in\mathbb{Z}[i]$ such that $a=bq +r$ and $N(r) < N(b)$ ...
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1answer
113 views

Number of invertible elements in quotient ring

I want to find an analog of Euler function $\varphi_{R}(a)$ that determines the number of invertible elements in the quotient ring $$R = \mathbb{F}_p[x]/(a), \text{ for } a \in \mathbb{F}_p[x]$$ ...
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2answers
76 views

Motivation of the definition of Euclidean Domain

We know the definition of Euclidean Domain is An Euclidean Domain is an Integral domain $(E,+,\ast )$ together with a function $v: E\setminus \{0\} \to \mathbb{N} \cup \{0\} $ such that (i) for all $...
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1answer
36 views

Division in $\mathbb{Z}[i]$ [duplicate]

I am studying the ring theory and I got some point like $(2+3i)$ divides $(-1+5i)$ in the Eucledian domain $\mathbb{Z}[i]$. I do not know that how it can be possible. Please let me know about this.
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1answer
36 views

Short inequality proof on $\mathbb{Z}[\sqrt{-2}]$ [closed]

Let $A=\mathbb{Z}[\sqrt{-2}]$ and let $a,b \in A,$ $b \neq0$. Set $\gamma := \dfrac{a}{b}$. I am struggling to show that there exists $c \in A$ such that $|c-\gamma|\leqslant \dfrac{{\sqrt{3}}}{2}$. ...
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1answer
21 views

Do boundedly finite homotopy groups of a decreasing collection of Euclidean domains stabilize?

Suppose $\{U_s \mid s \in (0,1)\}$ is a continuous nested collection of bounded Euclidean domains, i.e. each $U_t$, $t \in (0,1)$ is a bounded domain in $\mathbb{R}^n$ such that $U_s \subset U_t$ ...
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1answer
32 views

$m$ square-free integer such that $4\vert m-1\implies 2$ irreducible in $\mathbb{Z}[\sqrt{m}]$

Let $m$ square-free integer such that $4\vert m-1$. I need to prove that $2$ is irreducible in $\mathbb{Z}[\sqrt{m}]$. What I taught is to use the following theorem: LEMMA: Let $R$ a Euclidian ...
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1answer
70 views

Prove that in an Euclidean domain, if $d(a) = 1$ then $a$ is invertible or irreducible

Let $A$ be an Euclidean domain associated to a function $d:A-\{0\}\rightarrow \mathbb{N}$ and $a\in A$, $a\ne0$. Prove that if $d(a) = 1$, then $a$ is inversible or irreducible. I've tried using the ...
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2answers
117 views

$\mathbb{Z}[\sqrt{2}]$ is a euclidean domain

I need to prove that $\mathbb{Z}[\sqrt{2}]$ is a euclidean domain. I can use the function $$\phi:\mathbb{Z}[\sqrt{2}]\backslash\{0\}\rightarrow\mathbb{N} \\ \phi(a+b\sqrt{2})=|a^2-2b^2| $$ It was ...
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3answers
95 views

Why does Wolfram say domain of $y=\sqrt[3]{(x-1)^2(x+2)}$ is $x \in \mathcal{R} : x \ge 2$

Why does Wolfram say domain of $$y=\sqrt[3]{(x-1)^2(x+2)}$$ is $$x \in \mathcal{R} : x \ge 2$$ when I think is actually all real numbers because it's third root? Even google's quick answer is saying ...
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1answer
305 views

Show that every Euclidean ring is a principal ring

Question: Show that every Euclidean ring is a principal ring. My attempt: Let $(R, +, \cdot)$ be a Euclidean domain. Let $I \subseteq R$ be a nonzero ideal of $R$. Let $d \in I$ with $d \neq 0$ be ...
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1answer
101 views

Degree of the kernel of a module map $R^n\rightarrow R^n$ for an Euclidean domain $R$

Let $R$ be an Euclidean domain with the degree function $d$. Let $A\in R^{n\times n}$ be an $n\times n$-matrix with entries in $R$ such that det$(A)=0$. As a module map $A:R^n\rightarrow R^n$, there ...
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1answer
98 views

How many quotients/remainders can we get for a given Gaussian integer?

This is a question which arose1 from division of Gaussian integers. However, it seems to be basically a question about lattices in $\mathbb R^2$. We know that if we consider $\mathbb Z[i]=\{a+bi; a,b\...
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1answer
28 views

$A$ is a field iff $A[t]$ is euclidean.

I'm almost sure the question has already been asked but i don't know the english terminologies... I have in my lecture that : $A$ a ring. $A$ is a field iff $A[t]$ is principal. I'm anoyed ...
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1answer
52 views

Sufficient conditions for an element to be a unit in a euclidean domain

Let $(D, +, \cdot, \delta)$ be a euclidean domain. Prove that: a) An element $u \in D \setminus \{0\}$ is an unit if, and only if, $\delta(u) = \delta(1)$. b) If $u$ is an unit, then for each $x \in D ...
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1answer
89 views

It is given that $a_1=\gcd(a_{11}, a_{12} , \cdots , a_{mn})$. Then prove that after elementary row and column operation we get …

This is a question related to Smith Normal form I guess but I couldn't do the proof by my own using elementary row and column operation. So here is the problem: Suppose $$(a_{ij})= \begin{...
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1answer
22 views

Can anyone gives me an example of $V$ such that there exists at least one pair of elements $ a,c(\neq 0 ,1) \in D$ such that $V(a) > V(ac)$

$D$ is an Integral Domain. $V$ only satisfies the first Euclidean property, i.e. for all $\,a,b\in D\,$ if $\,b\neq 0\,$ then there are $\,q,r\in D\,$ such that $\, a = qb +r\,$ with either $r=0$ or $...
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1answer
53 views

Can anyone give me an example of an Euclidean Domain $D$ whose Euclidean Valuation at a point $a$: $v(a)$ is bigger than $\min \{v(ax) : x\in D\}$?

Can anyone give me an example of an Euclidean Domain $D$ whose Euclidean Valuation at a point $a$ , $v(a)$ is bigger than $\min \{v(ax) : x\in D\}$ ? I know that second condition for being Euclidean ...