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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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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 ...
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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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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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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 ...
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How to restrict only top range of ellipse function, and what is its domain?

I am trying to graph the function of an ellipse that is: $$1=\frac{x^2}{49}+\frac{(y+1)^2}{9}$$. I want to make the horizontal ellipse's range $y \leq 0.838$. So, when I also have to write the domain ...
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Rotman's “Advanced Modern Algebra”, exercise 3.97: a degree function of a Euclidean domain [duplicate]

Rotman defines a degree function of an integral domain $R$ as a function $\delta\colon R\setminus\{0\}\to\mathbb{N}$ so that $(1)$ for any $a,b \in R\setminus\{0\}$ we have $\delta(a) \leq \delta(ab)$...
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Visualizing euclidean rings

It is possible to see euclidean rings with lattices. For instance from : https://en.wikipedia.org/wiki/Gaussian_integer or from : https://en.wikipedia.org/wiki/Eisenstein_integer my question is ...
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Question about an Euclidean Domain (associates)

I know that $\mathbb{Z}$ is an ED with an euclidean function $\varphi\left(x\right)=\left|x\right| $. But in Euclidean Domain, Associates they say that ED iff Field. Isn't it contradictory? Is the ...
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Difficulty in showing $\mathbb Z[\frac{1+\sqrt{-7}}{2}]$ is euclidean domain using another way

This is not duplicate As I wanted to check why my argument is not working,Please have a look $\mathbb Z[\frac{1+\sqrt{-7}}{2}]$ is euclidean domain. My attempt: I had thought following proof for $...
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The set of the form $ m+n\sqrt{-3} $, $ m, n\in\mathbb Z $ or $ m, n $ are both halves of odd integers is a Euclidean domain

Let $ D $ be the set of complex numbers of the form $ m+n\sqrt{-3} $ where $ m $ and $ n $ are either both in $ \mathbb Z $ or are both halves of odd integers. Show that $ D $ is a Euclidean domain ...
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Associated elements in $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$

I got four elements and want to check whether these elements are associated or not. For $\alpha = \frac{1+\sqrt{-3}}{2}$ my elements are: $a_1 = 2 - \alpha = \frac{3 - \sqrt{-3}}{2}$ $a_2 = 1 - 2\...
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Gaussian Integers form an Euclidean Ring

A Ring $R$ is called euclidean if a map $f:R\backslash {0} \rightarrow \mathbb{N}$ exists with the following properties: For two elements $a,b \in R$ with $b\neq 0$ there exist $q,r\in R$ with: (i) $...
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In a Euclidean domain's division with remainders, is $r=0$ if $t$ divides $s$?

A Euclidean domain $D$ has a function $F$ from nonzero elements of $D$ to nonnegative integers that makes division with remainder possible in $D$: Let $D$ have elements $t$ and $s$, and $t$ is nonzero....
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Show that $\mathscr{O}_{\mathbb{Q}(\sqrt{-7})}$ is a UFD

It is known that the ring of integer is a Dedekind domain which means that it is a UFD iff it is a PID. Since $-7\equiv1$ mod $4$, we have that $\mathscr{O}_{\mathbb{Q}(\sqrt{-7})}=\mathbb{Z}\left[\...
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Solving the Diophantine equation $y^3 = 4x^2+4x+ 5$ for $x,y \in \mathbb{Z}$

I want to solve the Diophantine equation $y^3 = 4x^2+4x+ 5$ for $x,y \in \mathbb{Z}$. The right hand side factors as $(2x+1-2i)(2x+1+2i)$. Am I right that such a factorization can be found using the ...
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Can anyone make me understand in simple language why the second condition for being an Euclidean domain is superfluous?

Can anyone make me understand in simple language why the second condition for being an Euclidean domain is superfluous ? Why $v(a) \leq v(ab)$ is not needed? How we can deduce from the first one?
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How to prove that $\mathbb{Z}[(1+\sqrt{2})/2]$ is a Euclidean domain?

I know how to prove Euclidean domain for $\mathbb{Z}[\sqrt{2}]$ and $\mathbb{Z}[\sqrt{-2}]$, etc, but I am getting stuck on this notation. From what I've read, We let $R_d$ be the ring $\mathbb{Z}[(1+...
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Properties of rings of the form $\mathbb{Z}+\mathbb{Z}\sqrt{p}i+\mathbb{Z}\sqrt{q}j+\mathbb{Z}\sqrt{r}k$

It is well-known that $\mathbb{Z}+\mathbb{Z}\sqrt{5}i$ is not a unique factorization domain, since, for example, $6$ has two different factorizations into irreducibles: $(1+\sqrt{5}i)(1-\sqrt{5}i)=2\...
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Is there a “symmetric” way to use the Euclidean algorthm on $R[x,y]$ for a domain $R$?

Let $R$ be any integral domain, and $R[x,y]$ the ring of polynomials over $F$ in two variables. If we regard $R[x,y]$ as $\left(R[x]\right)[y]$, i.e. as polynomials in $y$ whose coefficients come ...
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Proving divisibility of a polynomial by a square of a polynomial.

I need to prove that a polynomial $f \left( x \right) \in \mathbb{Q} \left[ X \right]$ is divisible by a square of a polynomial iff $f$ and $f'$ have a greatest common divisor of positive degree. I ...
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Position of $\mathbb{Z}[\sqrt{-d}]$ in class hierarchy for all integers $d$

For any real number $d,$ define $$\mathbb{Z}[\sqrt{-d}]: = \{a+b\sqrt{-d}: a,b\in \mathbb{Z}\}.$$ It is well-known that $\mathbb{Z}[\sqrt{-2}]$ and $\mathbb{Z}[\sqrt{-1}]$ are ED but not fields, $$\...
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Programming the Euclidean algorithm for arbitrary Euclidean Domains in Maple

I have to implement the Euclidean algorithm for arbitrary domains in Maple. I know how to do this for integers: ...
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When $\mathbb{Q}[\sqrt{n}i]$ and $\mathbb{Z}[\sqrt{n}i]$ normal?

Denote $\mathbb{Q}[\sqrt{n}i]=\{a+b\sqrt{n}i | a,b \in \mathbb{Q}\} \subset \mathbb{C}$ and $\mathbb{Z}[\sqrt{n}i]=\{a+b\sqrt{n} | a,b \in \mathbb{Z}\} \subset \mathbb{C}$. The above two rings are ...
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Prove for every $i∈\Bbb Z∪{0}$ that $N(b^i)<N(b^{i+1})$ . [closed]

Let $D$ be a Euclidean domain with euclidean valuation $N$ . Suppose $b∈D$ is neither zero nor a unit. Prove for every $i∈\Bbb Z∪{0}$ that $N(b^i)<N(b^{i+1})$ . Euclidean valuation is a function: ...
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Prove that if $u\in \mathbb{Z}[\sqrt{2}]$ (Euclidean domain) is unit, then $v(u)=1$.

Prove that if $u\in \mathbb{Z}[\sqrt{2}]$ (Euclidean domain) is unit, then $v(u)=1$. Can someone let me know if this proof is okay? Let $\alpha$ be unit. Then there exists an element $\beta \in \...
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Prove $Z[\sqrt{7}]$ is euclidean domain [duplicate]

I am not able to prove it with euclidean evaluation $$ d(m+n\sqrt{7}) = |m^2 - 7 n^2 |$$. Is there any other way?
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Taylor's formula in two or more variables

I have a doubt about Taylor's formula in two or more variables:are these two scripts equivalent? if yes, why? $$f(\vec{x}) = f(\vec{x}^0) + \sum\limits_{i=1}^{m-1} \frac{1}{i!} \left( \dfrac{\...
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Norm-Euclidean Domains Division Algorithm

A norm-Euclidean domain is an Euclidean domain with respect to the norm function or the absolute value of the norm function. This implies that, for all $a,b \in R, b \neq 0$ where $R$ is a norm-...
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Given $a, b$ coprime integers, show that any factor of $a^2 - 2b^2$ is of the form $c^2 - 2d^2$

This is an exercise from Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory. I imagine the relevant ring here is $Z[\sqrt{2}]$, i.e., we can factor $a^2 - 2b^2$ into $(a + b\...
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GCD in arbitrary domain

Is there a domain where computing GCD of two elements is not trivial (i.e. Euclid's algorithm will not work)? AFAIK we can always use the Euclid's algorithm in a Euclidean Domain.
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Find the set of points where two functions are equal

Given the function: $$f(M, \vec{p}, d) = \sum_{i=1}^m log(\frac{\mathbb{I}(|\vec{M}_i - \vec{p}|_2<d)}{|\vec{M}_i - \vec{p}|_2})$$ Where $M \in \mathbb{R}^{m \times n}$ is a matrix, $\vec{p} \in \...
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2answers
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For a field $F$, what are the elements of $U(F[x])$?

Let $R$ be a Euclidean domain. An element $u \in R$ is said to be a unit if there is a $v \in R$ such that $uv = 1$. The set of units in $R$ is denoted by $U(R)$. Let $R$ be a commutative unital ...
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1answer
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Question regarding g.c.d of two polynomials

Let $f,g \in C[x,y]$ be two non-constant polynomials with no common factor. I want to prove that in the Euclidean domain $C(x)[y]$ the g.c.d of $f$ and $g$ is in $C(x)$ and it looks like I'm missing ...
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1answer
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The units of a Euclidean domain form a group

I'm unsure how to show how the units of a Euclidean domain form a group. I know: Let D be a domain. We shall call the function $\|x\|$ defined on the non-zero elements x of D with values in the ...
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1answer
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Prove ring of dyadic rationals is a Euclidean domain

This is a question from page 105, Vinberg - A course in Algebra: Prove that the ring $A$ of rational numbers of the form $ 2^{-n}m,~m \in \mathbb{Z},~n \in \mathbb{Z}_+ $, is a Euclidean domain. ...
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find the domain of this function ( [ ] is symbol of floor)

Find the domain of the function $f(x)=\frac{\log(3x-2x^2)}{\left \lfloor 2x-1 \right \rfloor^2 - 1}$ where $\lfloor \cdot \rfloor$ is the floor function.
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Least Euclidean function on $\mathbb{Z}\backslash\{0\}$

This is a problem in an Olympiad contest: Let $F$ be the set of all functions $f\colon \mathbb{Z} \backslash \{0\} \to \mathbb{N}$ with the property: if $a, b\in \mathbb{Z} \backslash \{0\}$ and $b$...
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Let $D$ be an domain of integrity. Show that $D$ is a domain of prime ideals if and only if there is an N-norm in D that satisfies:

i) If $a|b \Rightarrow N(a) \leq N(b)$ ii) If $a|b \ \text{and} \ N(a)=N(b) \Rightarrow b|a$ iii) If $a \nmid b \ \text{and} \ b \nmid a \Rightarrow \exists \ x,y \in D-\{0\}$ such that $ax+by \neq ...
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1answer
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Is $\mathbb{R}[x^2,x^3]=\{f=\sum_{i=0}^n a_ix^i\in\mathbb{R}[x]:a_1=0\}$ is Euclidean Domain?

Is $\mathbb{R}[x^2,x^3]=\{f=\sum_{i=0}^n a_ix^i\in\mathbb{R}[x]:a_1=0\}$ an Euclidean Domain ? My answer : I know that it is integral domain ,by theorem R is integral domain then $\Bbb R[x]$ ...
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Question about proof in Neukirch's Algebraic Number Theory

I was reading Proposition 2.2 in chapter I of Neukirch (page 6 in my edition), which states the following for an extension of rings $A\subseteq B$: (2.2) Proposition. Finitely many elements $b_1,\...
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“Let $R$ be an integral domain. Assuming that the Division Algorithm holds in $R[x]$, prove that $R$ is a field.” [duplicate]

Let $R$ be an integral domain. Assuming that the Division Algorithm holds in $R[x]$, prove that $R$ is a field. Can someone tell me if I am going about proving this correctly? My tactic was going to ...
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2answers
43 views

Why are irreducible elements prime in Euclidean domain?

I am trying to understand some notes from my algebra course, one of which says (it's basically a proof of that all irreducibles are prime in Euclidean domain) : ......Suppose $x = ab$ and $x$ does ...
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1answer
405 views

Definition of an Euclidean domain

From Wikipedia: Let $R$ be an integral domain. A Euclidean function on $R$ is a function $$f:R\setminus \{0\}\rightarrow\mathbb{Z}^+$$ satisfying the following fundamental division-with-...
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2answers
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Showing $(3 + \sqrt{3})$ is not a prime ideal in $\mathbb{Z}[\sqrt{3}]$

Let $I = (3+\sqrt{3})$ Looking at the field norm we note that $N(3 + \sqrt{3}) = 6$. We also know that $\mathbb{Z}[\sqrt{3}]$ is a Euclidean Domain. We want to find some $\alpha, \beta \in \mathbb{Z}...
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0answers
33 views

Norm-Euclidean fields

A finite field extension $\mathbb{F}/\mathbb{Q}$ is called Norm-Euclidean if the absolute norm function $\alpha \mapsto |\prod \sigma_i(\alpha)|$ induce a Euclidean norm on the integer ring $\mathcal{...
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Maximizing the Euclidean Distance of a Constrained Sum

Let $ {\bf a} \in \mathbb{C}^{N \times 1}$, where N is a positive integer. Denote the set of phasors ${\bf P}$ as ${\bf P} = \{1,\exp(j 2 \pi/M), \cdots, \exp(j 2 \pi (M-1)/M)\}$, where M is a ...
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0answers
68 views

Showing that $\mathbb{Z}[\phi]$ is a UFD, where $\phi^2+\phi+1=0$.

I would like to show that $\mathbb{Z}[\phi]$ is a UFD, where $\phi^2+\phi+1=0$. I started with a division with remainder to determine the irreducibles, but I could not finish that, and I do not know ...
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1answer
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Are 2x2 matrices on a Euclidean domain a Euclidean domain?

Let $M(R)$ be the set of 2x2 matrices with elements of ring $R$. $M(R)$ is obviously a ring also. But if $R$ is a Euclidean domain, is $M(R)$ also a eucliean domain? If it is, how do you perform ...
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2answers
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If the norm of $\alpha \in \mathbb{Z}[i]$ is a square, show that $\alpha$ is a square in $\mathbb{Z}[i]$

Suppose $\alpha \in \mathbb{Z}[i]$, $\alpha$ is not divisible by any integer, and $N(\alpha) = m^2, m \in \mathbb{Z}$. I want to show that $\alpha$ is a square in $\mathbb{Z}[i]$. I'm really lost ...
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3answers
879 views

Dividing one polynomial by another

How is this done? For example, how would one simplify the following? $$\frac{x^3-12x^2+0x-42}{x^2-2x+1}$$ I can do it with long division, but it never makes intuitive sense to me. Either an ...