query: Euclidean domain $\mathbb{Z}[\sqrt{d}]$
positive_passages.docid: 148804
positive_passages.text: Norm-Euclidean rings?
negative_passages.docid: 150677
negative_passages.text: What are the integers $n$ such that $\mathbb{Z}[\sqrt{n}]$ is integrally closed?
negative_passages.docid: 121986
negative_passages.text: Do the Laurent polynomials over $\mathbb{Z}$ form a principal ideal domain?
negative_passages.docid: 94579
negative_passages.text: $\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain, and its field of fractions is isomorphic to $\mathbb Q(t)$
negative_passages.docid: 80551
negative_passages.text: Find Units in $\mathbb{Z}[\sqrt{n}]$
negative_passages.docid: 81381
negative_passages.text: Why does the domain and range of $\sqrt x$ contain only positive real numbers?
negative_passages.docid: 138256
negative_passages.text: What's the difference between $\mathbb{Q}[\sqrt{-d}]$ and $\mathbb{Q}(\sqrt{-d})$?
negative_passages.docid: 95863
negative_passages.text: Describe the units in $\mathbb{Z}[\sqrt{-d}]$
negative_passages.docid: 62134
negative_passages.text: Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d $ roots
negative_passages.docid: 154845
negative_passages.text: Irreducible elements in  $\mathbb{Z}[\sqrt{-2}]$ and is it a Euclidean domain?
negative_passages.docid: 19822
negative_passages.text: For $D$ a GCD domain , let $a,b,x \in D \setminus \{0\}$ , then is it true that $\gcd (ax,bx)=x \cdot \gcd (a,b)$?
negative_passages.docid: 103550
negative_passages.text: Multiplicative Euclidean Function for an Euclidean Domain
negative_passages.docid: 110726
negative_passages.text: Example of a non-gcd domain
negative_passages.docid: 84017
negative_passages.text: Is $\sqrt[3]{2}$ contained in $\mathbb{Q}(\zeta_n)$?
negative_passages.docid: 19230
negative_passages.text: Prime/Maximal Ideals in $\mathbb{Z}[\sqrt d]$
negative_passages.docid: 58717
negative_passages.text: Why isn't $\mathbb Z [\sqrt{pq}]$ a factorial domain
negative_passages.docid: 141659
negative_passages.text: Every field is an integral domain