query: How many elements are there in $\mathbb{Z}[i]/I$ where $I= \langle 2+2i \rangle$
positive_passages.docid: 102721
positive_passages.text: Number of elements in the ring $\mathbb Z [i]/\langle 2+2i\rangle$
negative_passages.docid: 92517
negative_passages.text: Easy way to show that $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}[\sqrt[3]{2}]$
negative_passages.docid: 124438
negative_passages.text: Find $|(\mathbb{Z} \oplus \mathbb{Z}) / \langle(3,6)\rangle|$ and determine if its cyclic
negative_passages.docid: 78118
negative_passages.text: Algebraically, why is $\mathbb{Z}[i]/(i + 1)$ isomorphic to $\mathbb{Z}_{2}$?
negative_passages.docid: 68915
negative_passages.text: What are the units in $\mathbb{Z}[\root 3 \of 2]$?
negative_passages.docid: 89415
negative_passages.text: Number of homomorphisms from $(\mathbb Q^+ , \,\cdot\,)$ to $(\mathbb Z_n , +)$
negative_passages.docid: 73743
negative_passages.text: Number of elements in the quotient ring $\mathbb{Z}[X]/(X^2-3, 2X+4)$
negative_passages.docid: 7303
negative_passages.text: How many idempotent elements does the ring ${\bf Z}_n$ contain?
negative_passages.docid: 158315
negative_passages.text: Are $M_2= \langle(\sqrt{2}^{\sqrt{2}},\infty),\cdot \rangle$ , $M_1 = \langle(2,\infty),\cdot\rangle$ isomorphic structures?
negative_passages.docid: 120594
negative_passages.text: $\mathbb Z[i]/ \langle 1+2i \rangle \cong \mathbb Z_5$
negative_passages.docid: 89215
negative_passages.text: How to describe the quotient group $\mathbb{Z} \times \mathbb{Z} / \langle (1, 2) \rangle$
negative_passages.docid: 143069
negative_passages.text: What are the direct summands in $\mathbb{Z}^n$?
negative_passages.docid: 41069
negative_passages.text: Justifying $\,\Bbb{Z}[i]/\langle 2\rangle\cong \Bbb{Z}[X]/\langle X^2+1,2\rangle\cong \Bbb{F}_2[X]/\langle X^2+1\rangle$
negative_passages.docid: 146990
negative_passages.text: How to find the number of automorphisms on $\mathbb Z_2\times \mathbb Z_4$
negative_passages.docid: 76100
negative_passages.text: Showing that $\left\langle a,b \mid abab^{-1}\right\rangle \cong \pi_1(K) \cong \left\langle c,d \mid c^2 d^2 \right\rangle$
negative_passages.docid: 85752
negative_passages.text: Show that $M=\frac{\mathbb{Z}^3}{\langle (3,3,1),(2,2,2)\rangle}$ is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}_4$ as $\mathbb{Z}$-modules
negative_passages.docid: 15000
negative_passages.text: Show that number of elements in $(Z/nZ)^*$ is $\phi (n)$