query: Prove that $\mathbb{Z}[\sqrt{-2}]$ is a Euclidean domain
positive_passages.docid: 23048
positive_passages.text: The ring $\mathbb Z[\sqrt{-2}]= \{a+b\sqrt{-2} ; a\in \mathbb Z,b\in \mathbb Z \}$ has a Euclidean algorithm
negative_passages.docid: 75639
negative_passages.text: Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$
negative_passages.docid: 136855
negative_passages.text: Proof that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a PID
negative_passages.docid: 125604
negative_passages.text: Proof that $\mathbb{Z}$ has no zero divisors
negative_passages.docid: 93621
negative_passages.text: Prove that $\mathbb{Z}[\omega]$ is an integral domain
negative_passages.docid: 11259
negative_passages.text: Prove $R\times S$ is not an integral domain
negative_passages.docid: 25397
negative_passages.text: Prove $R \times R$ is NOT an integral domain
negative_passages.docid: 132193
negative_passages.text: Why $\Bbb Z$ = (integers) is an integral domain?
negative_passages.docid: 17859
negative_passages.text: Showing $F[x,y]/(ax^2+by^2-1)$ is a Dedekind domain
negative_passages.docid: 80551
negative_passages.text: Find Units in $\mathbb{Z}[\sqrt{n}]$
negative_passages.docid: 121845
negative_passages.text: Quotient of polynomials, PID but not Euclidean domain?
negative_passages.docid: 50930
negative_passages.text: Proving the continued fraction representation of $\sqrt{2}$
negative_passages.docid: 7011
negative_passages.text: Integral domain with fraction field equal to $\mathbb{R}$
negative_passages.docid: 47975
negative_passages.text: Closed cone in the euclidean space $\mathbb{R}^n$
negative_passages.docid: 146470
negative_passages.text: Show $\mathbb{Z}[\sqrt{6}]$ is a Euclidean domain
negative_passages.docid: 14158
negative_passages.text: Prove that $R$ is an integral domain $\Leftrightarrow$ $R[x]$ is an integral domain
negative_passages.docid: 46565
negative_passages.text: Show that $\mathbb{Q}(\sqrt{2})$ is a field.