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.

A Euclidean domain (also called a Euclidean ring) is a commutative ring that can be endowed with a Euclidean function, which allows a suitable generalization of the Euclidean division of the integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them (Bézout's identity). Moreover, every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.

It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but when an explicit algorithm for Euclidean division is known, one may use the Euclidean and extended Euclidean algorithms to compute greatest common divisors and the quantities in Bézout's identity. In particular, the existence of efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra.

So, given an integral domain R, it is often very useful to know whether R has a Euclidean function: if so, that implies that R is a PID. If there is no "obvious" Euclidean function, however, then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain.

Euclidean domains appear in the following chain of class inclusions:

commutative rings ⊃ integral domains ⊃ integrally closed domainsGCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domainsfieldsfinite fields

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Prove that the Gaussian Integer's ring is a Euclidean domain

I'm having some trouble proving that the Gaussian Integer's ring ($\mathbb{Z}[ i ]$) is an Euclidean domain. Here is what i've got so far. To be a Euclidean domain means that there is a defined application (often called norm) that verifies this two…
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The ring $\Bbb Z\left [\frac{-1+\sqrt{-19}}{2}\right ]$ is not a Euclidean domain

Let $\alpha = \frac{1+\sqrt{-19}}{2}$. Let $A = \mathbb Z[\alpha]$. Let's assume that we know that its invertibles are $\{1,-1\}$. During an exercise we proved that: Lemma: If $(D,g)$ is a Euclidean domain such that its invertibles are $\{1,-1\}$,…
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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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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 explanation of the long division algorithm or a new way…
Leif Metcalf
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$\mathbb{Z}[\sqrt{11}]$ is norm-euclidean

I'm trying to show that $\mathbb{Z}[\sqrt{11}]$ is Euclidean with respect to the function $a+b\sqrt{11} \mapsto|N(a+b\sqrt{11})| = | a^2 -11b^2|$ By multiplicativity, it suffices to show that $\forall x \in \mathbb{Q}(\sqrt{11}) \exists n \in…
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Euclidean domain $\mathbb{Z}[\sqrt{d}]$

I am trying to generalized, for which integral values of $d$, $\mathbb{Z}[\sqrt{d}] = \{ a + b\sqrt{d} \vert a,b\in\mathbb{Z}\}$ is an Euclidean domain? I am interested specially in positive integral values of $d$.
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Natural Euclidean Function Not Satisfying the $d$-inequality

Let me provide some background before I begin (although I feel as though it's hardly needed): Let $R$ be an integral domain. I call a function $d:R\setminus \{0\}\to\mathbb{N}\cup\{0\}$ a Euclidean function if for every $a,b\in R$, $a\ne0$,…
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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 $\alpha, \beta \in \mathbb{Z}[\sqrt{3}]$ s.t. $\alpha \cdot \beta = 3 +…
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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 always exists a kernel element $v\in R^n$ since…
Levent
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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 Euclidean domain $R$ has universal side…
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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 $$R = \mathbb{F}_p[x]/(a), \text{ for } a \in \mathbb{F}_p[x]$$ where $\mathbb{F}_p[x]$ is a Euclidian domain with…
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On a PID that is not an Euclidean domain

Let $\omega = \frac{1 + \sqrt{19}i}{2}$. The article here claims to prove that $\mathbb{Z}[\omega]$ is an example of a PID which is not a Euclidean domain. To prove that it is a PID, it takes an ideal $I$ and chooses a nonzero $b \in I$ as to…
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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 $$ D=\{x\in \mathbb R^d, \rho(x)<0\}, \quad \partial D=\{x\in \mathbb R^d, \rho(x)=0\}, $$ and $ x\in…
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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 $\mathbb Z[i]=\{a+bi; a,b\in\mathbb Z\}$ then we have a division with…
Martin Sleziak
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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 $\mathscr{O}_{\mathbb{Q}(\sqrt{-7})}=\mathbb{Z}\left[\frac{1+\sqrt{-7}}{2}\right]$. Now I read something…
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