query: {Proof Check} Let $D$ be a Euclidean Domain with Euclidean norm $\mu$. Then $\mu(1) = \mu(u) \iff u$ is a unit.
positive_passages.docid: 27167
positive_passages.text: Ideals in $F[x]$ and Euclidean domains are generated by any element of minimal degree
negative_passages.docid: 153411
negative_passages.text: If $|f|$ is constant, so is $f$ for $f$ analytic on a domain $D$.
negative_passages.docid: 90599
negative_passages.text: Trouble with proving $A$ is an integrally closed domain $\Rightarrow$ $A[t]$ is integrally closed domain
negative_passages.docid: 46746
negative_passages.text: How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?
negative_passages.docid: 157742
negative_passages.text: Prove that $U(n)$ is a manifold
negative_passages.docid: 96132
negative_passages.text: $R$ integral domain : $u\in R^*, a \text{ is prime} \iff au \text{ is prime}$
negative_passages.docid: 130073
negative_passages.text: $N(ab) \geq N(a)$ is not necessary for the ring to be euclidean
negative_passages.docid: 34147
negative_passages.text: $\|u\|\leq \|u+av\| \Longrightarrow \langle u,v\rangle=0$
negative_passages.docid: 122415
negative_passages.text: How to show every field is a Euclidean Domain.
negative_passages.docid: 236
negative_passages.text: Proof of general Euclid's Lemma in a UFD (by Euclidean algorithm?)
negative_passages.docid: 127248
negative_passages.text: Norm and unit sphere
negative_passages.docid: 82458
negative_passages.text: Homeomorphism between the unit disc and the unit square
negative_passages.docid: 82464
negative_passages.text: $U^*\otimes V$ versus $L(U,V)$ for infinite dimensional spaces
negative_passages.docid: 125465
negative_passages.text: Proving that $\sum \limits_{d^2|n}\ \mu (d)=\mu^2(n)$, where $\mu$ is the Möbius function.
negative_passages.docid: 114047
negative_passages.text: Metric $d(x,y)=\frac{|x-y|}{\sqrt{1+x^2}\sqrt{1+y^2}}$ on $\mathbb{R}$
negative_passages.docid: 106698
negative_passages.text: $\mathbb N$ when given the metric $d(m,n)=\frac{1}{m}-\frac{1}{n}$
negative_passages.docid: 154845
negative_passages.text: Irreducible elements in  $\mathbb{Z}[\sqrt{-2}]$ and is it a Euclidean domain?