# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 $A(\neq 0)$ ...