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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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Bachelor's Thesis - Euclidean Rings [on hold]

i will gratuate my bachelor's programme in applied mathematics and my supervisor teacher ask me to think about some topics related to Euclidean Domains. Can you give some topics please , something ...
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1answer
61 views

When the Euclidean map $ \delta $ is a ring homomorphism and satisfies $ \delta(a+b)\le\max(\delta(a), \delta(b)) $, then either a field or $ F[x] $

Let $ D $ be a Euclidean domain whose function $ \delta $ satisfies: (i) $ \delta(ab) = \delta(a)\delta(b) $, (ii) $ \delta(a+b)\le\max(\delta(a), \delta(b)) $. Show that either $ D $ ...
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1answer
25 views

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 ...
2
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1answer
34 views

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\...
4
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3answers
131 views

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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0answers
65 views

Prove $\mathbb{Z}[\sqrt{5}]$ and $\mathbb{Z}[\sqrt{6}]$ are Euclidean Domains

I would like to know how to prove $\mathbb{Z}[\sqrt{5}]$ and $\mathbb{Z}[\sqrt{6}]$. I don't know where to start, the method used for $\mathbb{Z}[\sqrt{2}]$ and $\mathbb{Z}[\sqrt{3}]$ doesn't work ...
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70 views

In a Euclidean domain's division with remainders, is $r=0$ if $t$ divides $s$?

Hi 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 ...
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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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1answer
83 views

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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1answer
62 views

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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25 views

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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2answers
32 views

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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0answers
23 views

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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1answer
21 views

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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1answer
41 views

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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1answer
58 views

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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1answer
39 views

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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0answers
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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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0answers
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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{\...
2
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1answer
52 views

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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1answer
81 views

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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5answers
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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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1answer
63 views

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
30 views

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
30 views

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
29 views

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
46 views

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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1answer
31 views

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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1answer
51 views

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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0answers
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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
43 views

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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1answer
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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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0answers
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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
35 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
243 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
24 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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0answers
16 views

Howto get scalar multiplication elementary matrix out of row swap operations in a Euclidean domain

My original post: Prove that, over a Euclidean domain $R, I + cE_{ij}$ generate $SL_n(R)$. Following the comments made, it now suffices to show that we can get the one-row scalar multiplication ...
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0answers
27 views

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
52 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
65 views

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 ...
2
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2answers
39 views

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 ...
12
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3answers
872 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 ...
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3answers
316 views

$\mathbb{Z}[i] / (2+3i)$ has 13 elements

I am self studying algebra, and have come across this exercise: Show that $\mathbb{Z}[i]/(2+3i)$ has 13 elements and is a field. My proof: Since Z[i] is a euclidian domain with euclidian norm ...
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1answer
32 views

Comparing orders induced by euclidean function and divisibility in euclidean domain

Let $(R,+,\cdot ,\Phi)$ be a euclidean domain where $\Phi$ denotes the Euclidean function, that is a function $R \setminus \lbrace 0 \rbrace \rightarrow \mathbb N$ such that : $\forall a \in R \quad \...
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1answer
35 views

Ring of remainders definition

My notes on Euclidean domains state a notion of ring of remainders that I should connect with the quotient by certain ideal but I don't see quite how the connection is done. Given an euclidean ...
1
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2answers
89 views

Looking for more direct proof of non-existence of “Euclidean algorithm norm” for $\mathbb{Z}[\sqrt{-5}]$

The textbook I'm reading, Integers, Polynomials, and Rings, by Ronald Irving, states on p. 246: A ring $R$ is called a Euclidean ring if it satisfies the following three properties: A. There ...
0
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0answers
28 views

$R$ euclidean domain, show $\tilde{N}(a)=\min_{0\neq x\in\left<a\right>}N(x)$ is a norm [duplicate]

Let $R$ be an euclidean domain with a norm $N$. We define $\tilde{N}(a) = \min_{0\neq x\in\left<a\right>}N(x)$. Show that $\forall a,b\neq 0\quad \tilde{N}(ab) \geq \tilde{N}(a)$ $\tilde{N}(a)...