# 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.

252 questions
Filter by
Sorted by
Tagged with
17 views

• 105
76 views

### Uniqueness of Division in Euclidean domain

I was reading this old paper which asserts that if $E$ is an euclidean domain with unique division we have $E\approx \mathbb{F}$ or $E\approx \mathbb{F}[X]$ where $\mathbb{F}$ is a field. The author ...
• 2,374
4k views

49 views

### Is the quotient by a unit in an euclidean domain unique?

I've seen some conditions for quotients and reminders to be unique, though they are not in general. However I haven't been able to find a counter example for the case when "dividing" by a ...
• 2,772
129 views

### If $p\in\mathbb{Z}$ is a prime and $p\equiv 1$ mod $4$, show that the quotient ring $\mathbb{Z}[i]/(p)$ has order $p^2$.

The Problem: Let $p\in\mathbb{Z}$ be a prime with $p\equiv 1$ mod $4$. Show that $\mathbb{Z}[i]/(p)$ has order $p^2$. Source: Abstract Algebra $\mathit{3^{rd}}$ edition by Dummit and Foote. My Attempt:...
• 1,467
731 views

### Show that the quadratic integer ring $\mathcal{O}=\{a+b\frac{1+\sqrt{-3}}{2}|a, b\in\mathbb{Z}\}$ is an Euclidean Domain.

The Problem: Let $F=\mathbb{Q}(\sqrt{-3})$ be a quadratic field with associated integer ring $\mathcal{O}=\mathbb{Z}[\omega]=\{a+b\omega\mid a, b\in\mathbb{Z}\}$ where $\omega=\frac{1+\sqrt{-3}}{2}$, ...
• 1,467
18 views

### Suppose $R$ is an integral domain, $a, b\in R$ are nonzero. Suppose $x$ is nonzero, then $x\in(a, b)$ IFF $x=sa-tb$ for some $s, t\in R$. [duplicate]

The Problem: Suppose $R$ is an integral domain, $a, b\in R$ are nonzero. Suppose $x$ is nonzero, then $x\in(a, b)$ IFF $x=sa-tb$ for some $s, t\in R$. My Attempt: Certainly this is true if $R$ is an ...
• 1,467
41 views

### Let $p\in\mathbb{N}$ be prime. Prove that p is irreducible in $\mathbb{Z}[i]$ if and only if $p\equiv3$ $(mod$ $4)$. [duplicate]

I have already proven one direction, which is $p\equiv3$ $(mod$ $4)$ implies $p$ irreducible. Now, stating that $p$ is irreducible I can't get to the conclusion that $p\equiv3$ $(mod$ $4)$. I have ...
311 views

### Proof that every element of a Euclidean domain has a $\gcd$ [duplicate]

I am required to prove that every element $a$ and $b$ of a Euclidean domain $E$ has a greatest common divisor. Here's my approach. Let $D=\{ax + by: a,b,x,y \in E \}$ We claim that $D$ is an ideal of ...
1 vote
108 views

### For an euclidean domain $R$, there is $r\in R \setminus R^×$ with surjective projection $p:R^×\cup \{0\} \rightarrow R/Rr$ [universal side divisors]

I think I should pick $r$ such that the value of $f(r)$ (where $f$ is an euclidean function) is minimal, but I’m not sure what to do next.
• 355
76 views

### Showing $\mathbb{Z}[\alpha] / (2)$ and $\mathbb{Z}[\alpha] / (3)$ are fields

I'm doing an elementary commutative algebra / number theory exercise, the goal is to show that $A=\mathbb{Z}[\alpha]$ ($\alpha = \frac{1+i\sqrt{19}}{2}$) is a principal ideal domain which is non ...
• 1,048
69 views

166 views

• 2,295
50 views

### Let $D = \mathbb{Z}[i\sqrt{3}]$ and $v(a +bi\sqrt{3}) = a^2 + 3b^2$ then show that $v$ is not Euclidean evaluation map on $D$ [duplicate]

EDIT: somebody tried to close this question saying that this question similar to this. But I think my question is different,I want to show that $\mathbb{Z}[i\sqrt{3}]$ doesn't satisfy the basic ...
• 293
146 views

• 293