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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21 views

Is it necessary for an Euclidean domain to satisfies Triangle inequality

I am wondering whether there always exist a Euclidean function f that satisfies $$f(a+b) \le f(a)+f(b).$$ In all the ED I know, it seems to be true. For example, integers, Gaussian integers, all the ...
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24 views

Prove that $\forall z \in \Bbb C, \exists x \in \Bbb Z[j], |z-x|\leq \frac{\sqrt3}{2}$

Need help proving that $\forall z \in \Bbb C, \exists x \in \Bbb Z[j], |z-x|\leq \frac{\sqrt3}{2}$ \ where $\Bbb Z[j]=\{a+jb / a,b \in \Bbb Z\}$ and $j=e^{\frac{i2\pi}{3}}$. I thought of using the ...
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2answers
117 views

What is the structure of $\mathbb Z[i]/\mathfrak p$ where $\mathfrak p$ is a prime.

Initially, I was trying to prove both the isomorphism of $\mathbb Z[i]/\mathfrak p\cong \mathbb{Z}[x]/(p,x^2+1)\cong \mathbb{F}_p[x]/(x^2+1)$, where $\mathfrak p$ is a prime in $\mathbb Z[i]$ for some ...
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0answers
167 views

Stillwell, Elements of NT, exercise 11.3.4. Proof check

I'm reading Stillwell's marvelous book Elements of number theory. This is not a homework question, I'm studying it for myself. I solved the question below and want to ask if my solution is correct. ...
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0answers
19 views

High model performance degradation on use of cosine as distance function in machine learning compare to Euclidean

I was working on classification of images based on similarity using machine learning. I am experimenting with both Euclidean and cosine distance. I found that as the no of classes grows, cosine as ...
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0answers
29 views

Primes and irreducible elements in Euclidean domain

I was studying Algebra (partially from I.N. Hertstein book). In it, an element is defined as prime $\pi$, if $\pi = ab \implies \text{a or b is unit.}$ However, this is the definition for irreducible ...
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2answers
64 views

A doubt in the proof of $\mathbb{Z} + \mathbb{Z}\sqrt m$ is not Euclidean with respect to $\phi_m$

I am reading 'Introductory Algebraic Number Theory' written by Saban Alaca and Kenneth S. Williams. Theorem 2.3.1 says that Let m be a positive squarefree integer. If there exist distinct odd primes ...
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45 views

Implications between $K$ field and $K[x]$ PID and ED

I know that if $K$ is a field, then $K[x]$ is an ED (Euclidean Domain), and that if $K[x]$ is a PID (Principal Ideal Domain) then $K$ is a field. Now, I can say that if $K[x]$ is an ED, then it is a ...
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1answer
79 views

GCD of rational and irrational numbers

We know that in every PID, for every two elements, there is one GCD (up to being associate). Now, since the set of real numbers is a field and consequently PID, this theorem holds for them. Then, what'...
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1answer
55 views

Polynomial division of polynomials in two variables

I have the following problem: Let's say I have two polynomials $f,g\in R[X,Y]$ for some domain $R$, and I want to check if $g$ divides $f$. If both $f,g$ were in $R[X]$ then I would do polynomial ...
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Of those domains that were proven to be Euclidean but are not norm-Euclidean, can we really perform Euclidean algorithm in them?

There are domains that are not norm-Euclidean but were proven to be Euclidean using Motzkin's transfinite Euclidean function. In this answer (let's call this answer 1), https://math.stackexchange.com/...
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Does $\forall a\neq 0,N(a)>N(0)$ imply $N$ is a Euclidean function? [duplicate]

Let $A$ be an integral domain. A Euclidean function on $A$ is often defined as a map $N:A\to\Bbb{ N}$ such that the following two properties hold: (1) For all $a,b\in A, b\neq 0$, we have $N(a)\leq N(...
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Compactness and Cantor's Criterion [closed]

Cantor's Criterion: If $\left\{C_{k}\right\}$ is a nested sequence of closed bounded, nonempty subsets of $\mathbb{R}^{n}$, then $$ \bigcap_{k=1}^{\infty} C_{k} \neq \emptyset $$ Furthermore, if $\lim ...
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1answer
28 views

Angle between arbitrary rectangle and horizontal plane

This seems to me like a fairly simple problem but I'm constantly re-thinking it because something seems wrong about how I'm solving it. I feel like there may be perhaps an easier solution or something ...
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4answers
131 views

Finding the units of $\mathbb{Z}\left[\frac{1+\sqrt{-19}}{2}\right]$

I was trying to comprehend a simple exercise from my elementary number theory class. Let $\theta=\frac{1+\sqrt{-19}}{2}$, and let $R$ be the ring $\mathbb{Z}[\theta] .$ Show that the units of $R$ are ...
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31 views

Euclidean functions on polynomial rings over fields

I was wondering whether there were Euclidean functions on F[X] (F is a field) that aren't the standard degree one. I know that F[X] is a uniquely Euclidean domain but I don't know how to use that (if ...
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1answer
57 views

When is $R[x]$ a Euclidean domain?

It is well-known that $R[x]$ is a PID iff $R$ is a field. Is there a necessary and sufficient condition on $R$ for $R[x]$ to be a Euclidean domain?
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$\Bbb Z\left [\frac{-1+\sqrt{-19}}{2}\right ]$ is not a Euclidean domain

Definition: A universal side divisor, is an element $s\in R\setminus R^\times$ such that for every $x\in R$ either $s\mid x$, or there is some unit $u\in R^\times$ such that $s\mid x+u$. Fact: A ...
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115 views

Show that $\mathbb{Z}[\sqrt{14}]$ is not an Euclidean domain with regard to this norm.

For every $a+b\sqrt{14}∈\mathbb{Z}[\sqrt{14}]$ define the norm function by: $N(a+b\sqrt{14})=|a^2-14b^2|$ Show that $\mathbb{Z}[\sqrt{14}]$ is not an Euclidean domain with regard to this norm. My ...
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1answer
67 views

Regarding the Tensor Product of Quotient Rings of a Euclidean Domain

Consider a field $k$ of characteristic $0$ and a positive integer $n.$ In the proof of Theorem 4.19 of Polytopes, Rings, and K-Theory by Bruns and Gubeladze, it is stated that $k[x] / (x^n - 1)$ is a ...
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1answer
172 views

On $C^1$ convex domain

Let $D$ be a $C^1$ domain of $\mathbb{R}^d$. Then we know that there exists a $C^1$ function $\rho:\mathbb R^d\rightarrow \mathbb R$ such that $$ D=\{x\in \mathbb R^d, \rho(x)<0\}, \quad \partial D=...
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116 views

Examples of Euclidean domains

An integral domain $E$ is called Euclidean domain if there exist a function (often called as norm) $f$ from non zero elements of $E$ to the non negative integers such that $f(a) \le f(ab)$ for all $a,...
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49 views

Proving that $aR + bR = \gcd(a, b)R.$ [duplicate]

How can I prove this: In a Euclidean domain $R,$ we have that $aR + bR = \gcd(a, b)R$? Here are my thoughts: I know that if $d = \gcd(a, b),$ then it can be written as a linear combination of $a,b$ ...
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1answer
201 views

An application of Brouwer fixed point Theorem

Let $\zeta:\mathbb{R}\to [0,+\infty)$ be a continuous non-negative function such that $\zeta(0)=0$ and $\tau\mapsto \zeta(\tau)\tau$ is a non-decreasing differentiable function whose derivative is ...
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0answers
35 views

Properties of size function in a general Euclidean domain [duplicate]

In ring theory a given ring $R$ is called a Euclidean domain if there exists a function $\sigma:R -\{0\}\rightarrow \{0,1,2,3...\} $ which satisfies the division algorithm i.e. $ $ if $a,b \in R$ ...
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80 views

Proving Eisenstein integers form a Euclidean domain

Let $\omega=\frac{-1+i\sqrt3}{2}=e^{\frac{2\pi i}{3}}$ and define the ring $R = \mathbb{Z}[\omega] = \{a+b\omega\mid a,b\in\mathbb{Z}\}$ with Euclidean function $\phi(a+b\omega)=a^2-ab+b^2$ and show ...
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2answers
43 views

Find a generator of the ideal generated by $3$ and $2-2\sqrt{-2}$ in ${\mathbb{Z}\left[\sqrt{-2}\right]}$

Exercise: Find a generator of the ideal generated by $3$ and $2-2\sqrt{-2}$ in ${\mathbb{Z}\left[\sqrt{-2}\right]}$. I know that such a generator exists since $\mathbb{Z}\left[\sqrt{-2}\right]$ is a ...
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87 views

Is $\mathbb{Z}[i,\varphi]$ a Euclidean domain?

Here $\varphi=\frac{1+\sqrt{5}}{2}$. It's true that $\mathbb{Z}[\varphi]=\mathcal{O}_{\mathbb{Q}(\sqrt{5})}$ is Euclidean since $\mathbb{Q}(\sqrt{5})$ is norm Euclidean, and I've read that $A=\mathbb{...
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1answer
60 views

Euclidean division in quotients of polynomial rings

I know that given a field $\mathbb{K}$, the one variable polynomial ring $\mathbb{K}[x]$ is an euclidean domain. This helps to figure out how the quotient $\dfrac{\mathbb{K}[x]}{(f(x))}$ (where $f(x) \...
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1answer
178 views

Show that the ring of formal power series is a Euclidean domain.

Here is the question I want to solve part $(c)$ of it: A commutative ring $R$ is local if it has a unique maximal ideal $\mathfrak{m}.$ In this case, we say $(R, \mathfrak{m})$ is a local ring. For ...
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0answers
28 views

How do I find $\displaystyle {\inf\limits_{n,m} \dfrac {\deg (m)} {\deg (r)}}\ ?$

Given $n,m \in \Bbb Z[i],$ we have $n = qm + r,\ r \in \Bbb Z[i].$ Then what is the value of $\displaystyle {\inf\limits_{n,m} \dfrac {\deg (m)} {\deg (r)}}\ ?$ How do I solve this question? Any help ...
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2answers
43 views

How many solutions there are for $x \in Z_6$ in the following equations?

How many solutions there are for $x \in Z_6$ in the following equations? $x\cdot[3]_6 = [4]_6$ $x\cdot[5]_6 = [4]_6$ $x\cdot[5]_6 = [3]_6$ $x\cdot[3]_6 = [3]_6$ I can find the number of solutions ...
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55 views

Does every pair of non-zero elements of $Z[\sqrt{3}]$ has a gcd?

I have a question as follows: Does every pair of non-zero elements of $Z[\sqrt{3}]$ has a gcd? Now, since $Z[\sqrt{3}]$ is a Euclidean domain as I saw a proof in one of the already answered questions ...
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1answer
61 views

Extending Euclidean Valuation (algebra chapter 0)

Exercise 3.2.15 in Aluffi's Algebra: Chapter 0 goes as follows: Given a Euclidean Domain $R$ with valuation $v$, show that there exists a Euclidean valuation $\overline{v}$ on $R$ such that $\...
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120 views

Algorithmic Euclidean domains.

I've been studying about the division algorithm and found out that an euclidean domain may or may not have a division algorithm, but I've not found any examples of such domains. Are they rare? Can we ...
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2answers
82 views

About a subset of $\mathbb Q[x]$ of polynomials $f$ such that $f(n)=f(-n)$ for every $n$ in $\mathbb N$

Let $A=\{f \in \mathbb Q[x] : f(n)=f(-n)$, for every $n \in \mathbb N\}$. Show that $A$ is a subring of $\mathbb Q[x]$. $A$ is a Euclidean Domain. For every $f \in A$ we have $f(r)=f(-r)$, for every $...
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3answers
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Problem in ordering of the $\mathbb{C}$

The question given in Rudin is that , prove that there doesn't exist an order such that $\mathbb{C} $turns into an ordered field. An order on S is a relation with the following properties (i) If x,y ...
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1answer
80 views

For this broad definition of Euclidean domain, is there a non-trivial example with finite set of norms?

This is a follow-up to the accepted answer to the question Euclidean mapping question. We call an integral domain $R$ Euclidean if there exists a function (called a "norm") $N: R\setminus\{0\...
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1answer
78 views

Euclidean mapping question

Give $A$ is Euclidean domain and $\delta: A\setminus \left\{0\right\} \to \mathbb{N}$ is Euclidean mapping. I have prove that "$A$ is a field if and only if $\delta$ is constant". So, I ...
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1answer
112 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
45 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
214 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
441 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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0answers
210 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
33 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
18 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
49 views

Conditions for a Euclidean domain to be a field or a polynomial ring over a field

I am having trouble proving the following. Let $R$ be a 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{max}(...
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1answer
30 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
36 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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56 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 $...