query: Let $R=\mathbb{Z}[i]=\{a+bi\mid a,b\in\mathbb{Z}\}$, where $i^2 =−1$, and let $I=(3+i)$. Find the exact number of elements in $R/I$.
positive_passages.docid: 54585
positive_passages.text: Order of some quotient ring of Gaussian integers
negative_passages.docid: 109764
negative_passages.text: Find the cardinality of the quotient of $\mathbb R$ in respect to R
negative_passages.docid: 107821
negative_passages.text: The ring of integers of $\mathbf{Q}[i]$
negative_passages.docid: 111824
negative_passages.text: Sum of nilpotent and element is a unit in ring?
negative_passages.docid: 50412
negative_passages.text: Prime elements in $\mathbb{Z}/n\mathbb{Z}$
negative_passages.docid: 24203
negative_passages.text: Finding Zero Divisors and Units of Rings $\mathbb{Z}_n \times \mathbb{Z}_m$
negative_passages.docid: 158182
negative_passages.text: Let $R = \mathbb{Z} + x\mathbb{Q}[x]$. Find all the irreducibles in $R$.
negative_passages.docid: 14191
negative_passages.text: How many elements does the ring $ℤ[X]/(X^2-3,2X-4)$ have? Describe the structure of this ring.
negative_passages.docid: 35691
negative_passages.text: $\mathbb Z[\sqrt 3]$ contains infinitely many units
negative_passages.docid: 66313
negative_passages.text: $\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$ if $(a,b)=1$
negative_passages.docid: 42875
negative_passages.text: The prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $.
negative_passages.docid: 154305
negative_passages.text: Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
negative_passages.docid: 118299
negative_passages.text: Show that $(\mathbb{Z}[x]/(x^{n+1}))^{\times}\cong \mathbb{Z}/2\mathbb{Z}\times\Pi_{i=1}^n\mathbb{Z}$
negative_passages.docid: 53575
negative_passages.text: Describe $R=\mathbb{Z}[X]/(X^2-3,2X+4)$
negative_passages.docid: 140144
negative_passages.text: Can you find an element of order 10 in $\mathbb{Z}_{31}^{*}$
negative_passages.docid: 21840
negative_passages.text: Zero divisors and units for the ring $\mathbb{Z}/n\mathbb{Z}$ with integer $n$
negative_passages.docid: 69582
negative_passages.text: Is it true that: $a+bi$ is prime in $\mathbb{Z}[i]$ if and only if $a^2+b^2$ is prime in $\mathbb{Z}$