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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Ring of integer for the case D=-19 is not a Euclidean domain [duplicate]
I'm solving the following problem. Unfortunately, I don't know any source of the problem because it's a question from another school.
Let $R=\{a+b\alpha|a, b\in \Bbb{Z}, \alpha=\frac{1+\sqrt{-19}}{2}\}...
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In $\Bbb Z[i]$ find $\gcd(2-7i,2+11i).$ Also find $x,y\in \Bbb Z[i]$ such that $(2-7i)x+(2+11i)y=\gcd(2-7i,2+11i).$ [duplicate]
In $\Bbb Z[i]$ find $\gcd(2-7i,2+11i).$ Also find $x,y\in \Bbb Z[i]$ such that $(2-7i)x+(2+11i)y=\gcd(2-7i,2+11i).$
I tried solving the problem as follows:
Let $d=\gcd(2-7i,2+11i).$ So, $$d|2-7i,d|2+...
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An Element in Euclidean Domain is Prime if and only if Universal Side Divisor. (Is This True?)
Claim. (Is this true?) An element $u\in \text{Euclidean Domain }R$ is prime if and only if it denotes a universal side divisor.
Proof Attempt. Let $u\in \text{Euclidean Domain }R$ denote some ...
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Understanding the Motivation Behind Euclidean Domains
To provide explicit questions:
What is the Euclidean Domain supposed to be a generalization of (with respect to its history)? In other words, was there a ring which motivated the initial construction ...
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Show that if $R$ is an integral domain and $f(x)$ is a unit in the polynomial ring $R[x]$, then $f(x) \in R$ [duplicate]
Show that if $R$ is an integral domain and $f(x)$ is a unit in the polynomial ring $R[x]$, then $f(x) \in R$.
Proceed by contraposition. Suppose $R$ is an integral domain, $f(x)$ is a unit in $R[x]$, ...
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Euclidean domain question involving euclidean function and associates [duplicate]
Given R an euclidean domain with euclidean function f:R - {0} -> N, a and b non zero elements in R such that f(a) = f(b) and a divides b. Prove that a and b are associated.
What I got until now:
b =...
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Calculating/enumerating the euclidean projection between two spheres
Let
$P \subset \mathbb{R}^n$ be a finite set of $n$-dimensional euclidean points;
$r_p \in \mathbb{R}^{+}$ be the radius of the sphere centered at $p \in P$; And
$c_p = \{ x \in \mathbb{R}^n : \|x - ...
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Construction of connected set in expanded complex plane
I'm trying to prove a theorem(I'm not sure whether it's right)
$X\cup\{\infty\}$ is connected in $\overline{C}$(expanded complex plane) if and only if all connected components of $X$ in $C$ are ...
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polynomials with leading coefficient unit are cancellable?
Let $R$ be a commutative ring with unit.
The general statement of Euclidean Division in $R[x]$ is the following.
If $f, g \in R[x]$ and $g \neq 0$ with the leading coefficient of $g$ being a unit, ...
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If $a$ and $b$ are coprimes in an euclidean domain, then $a^n$ and $b$ are coprimes. [duplicate]
Let $a,b\in\mathbb{E}$ where $\mathbb{E}$ is an euclidean domain. Prove that if $a$ and $b$ are coprimes, then $a^n$ and $b$ are coprimes for all $n\in\mathbb{N}$.
I think the proof is by induction on ...
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Factorizaton in an Euclidean ring
I have a doubt concerning Lemma 3.7.4 from Topics in Algebra by I. N. Herstein.
The statement of the Lemma is:
Let $R$ be a Euclidean ring. Then every element in $R$ is either a unit in $R$ or can be ...
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showing that $\mathbb Z [\sqrt{14}]$ is not euclidean regarding usual norm using only 2 and 1 + $\sqrt{14}$.
I'm trying to prove $\mathbb Z [\sqrt{14}]$ is not euclidean regarding the Norm $N(a+b\sqrt{14})=|a^2-14b^2|$ using $2$ and $1+\sqrt{14}$.
I know how to prove it for the Gaussian integer case and I ...
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Need help in understanding a proof of the fact: The set of Gaussian integers is a euclidean domain.
Let $J[i]$ denote the set of Gaussian integers. Show that $J[i]$ is a Euclidean ring.
The proof given is as follows:
In order to show this we must first introduce a function $d (x)$ defined for every ...
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Problem in understanding the unique factorization theorem for Euclidean Rings.
Unique Factorisation Theorem: Let $R$ be a Euclidean ring and $a\neq 0$ non-unit in $R.$ Suppose that $a =\pi_1\pi_2\cdots\pi_n=\pi_1'\pi_2'\cdots\pi_m'.$ where the $\pi_i$ and $\pi_j'$ are prime ...
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If $f:R\to S$ is a surjective homomorphism of integral domains, $p$ is irreducible in $R,$ and $f(p)\neq 0_R$ is $f(p)$ irreducible in $S?$
Background:
Exercise 24: If $f:R\to S$ is a surjective homomorphism of integral domains, $p$ is irreducible in $R,$ and $f(p)\neq 0_R$ is $f(p)$ irreducible in $S?$
Questions:
My guess is no. If we ...
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