# 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.

105 questions
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $R$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $\mathbb{Z}[i] / (2+3i)$ 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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