query: Why is the ring of integers in $\mathbb{Q}(\sqrt{m}) = \mathbb{Z}[\frac{1+\sqrt{m}}{2}]$ when $m \equiv 1\pmod{4}$?
positive_passages.docid: 19473
positive_passages.text: Why is quadratic integer ring defined in that way?
negative_passages.docid: 86807
negative_passages.text: The units of $\mathbb Z[\sqrt{2}]$
negative_passages.docid: 27704
negative_passages.text: Cardinality of quotient ring $\mathbb{Z_6}[X]/(2X+4)$
negative_passages.docid: 19310
negative_passages.text: Identify the ring $\mathbb{Z}[x]/(2x)$
negative_passages.docid: 55050
negative_passages.text: How to show that $\mathbb{Q}(i,\sqrt[4]{1+i} )$ is not contained in any cyclotomic extension of $\mathbb{Q}(i)$?
negative_passages.docid: 147871
negative_passages.text: How do the prime ideals of $\mathbb{Z}_{k}$ look like?
negative_passages.docid: 144789
negative_passages.text: identify the ring $\mathbb{Z}[x]/(2x-1)$
negative_passages.docid: 62183
negative_passages.text: Find the quotient field of $\mathbb{Z}[\sqrt {2}]$
negative_passages.docid: 87682
negative_passages.text: Show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$
negative_passages.docid: 151423
negative_passages.text: Is there a nice description of the field of fractions of the ring of polynomials with integer coefficients?
negative_passages.docid: 127814
negative_passages.text: Number of elements in the quotient ring $\mathbb{Z}_6 [x]/\langle 2x +4\rangle$
negative_passages.docid: 92583
negative_passages.text: Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$
negative_passages.docid: 55900
negative_passages.text: Is there a ring homomorphism $M_2(\mathbb Z)\to \mathbb Z$?
negative_passages.docid: 154845
negative_passages.text: Irreducible elements in  $\mathbb{Z}[\sqrt{-2}]$ and is it a Euclidean domain?
negative_passages.docid: 63478
negative_passages.text: Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?
negative_passages.docid: 68915
negative_passages.text: What are the units in $\mathbb{Z}[\root 3 \of 2]$?
negative_passages.docid: 133238
negative_passages.text: Why does $\mathrm{Tor}_0^R(M,N)\cong M\otimes_R N$?