query: Show that $ | \mathbb Z[\sqrt{-1}]/(a+b\sqrt{-1}) |=a^2+b^2 $
positive_passages.docid: 31263
positive_passages.text: Quotient rings of Gaussian integers
negative_passages.docid: 34693
negative_passages.text: Prove $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$ if $E^{\mathbb P}[X]=\mathbb P(X>0)=1$ and $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $
negative_passages.docid: 136304
negative_passages.text: Prove: $2^b-1\mid{}2^a-1 \iff b\mid{}a$
negative_passages.docid: 90276
negative_passages.text: Is $\sqrt{2}\in{\Bbb Z}[\sqrt{2}+\sqrt{3}]$ true?
negative_passages.docid: 85525
negative_passages.text: Prove that $x^3+x+1$ splits in $Z_2[x]/(x^3+x+1)$
negative_passages.docid: 38489
negative_passages.text: Degree of the extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$
negative_passages.docid: 25907
negative_passages.text: Show $a\Bbb Z+b\Bbb Z = \gcd(a,b)\Bbb Z$
negative_passages.docid: 98577
negative_passages.text: Show that the set $\mathbb{Q}(\sqrt{p})=\{a+b\sqrt{p}; a,b,p\in\mathbb{Q},\sqrt{p}\notin \mathbb{Q}\}$ is a field
negative_passages.docid: 87682
negative_passages.text: Show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$
negative_passages.docid: 156356
negative_passages.text: Proof that $x^n+\frac{1}{x^n}\in \mathbb Z$ by complete induction
negative_passages.docid: 52800
negative_passages.text: Show that $z^5-1=(z-1)(z^2+2z\cos{\pi \over 5}+1)(z^2-2z\cos{2\pi \over 5}+1)$
negative_passages.docid: 98496
negative_passages.text: Let $a,b \in \mathbb{Q^{+}}$ then $\sqrt{a} + \sqrt{b} \in \mathbb{Q} $ iff $\sqrt{a} \in \mathbb{Q}$ and $\sqrt{b} \in \mathbb{Q}$
negative_passages.docid: 16774
negative_passages.text: If $x,y,z\in\mathbb R\setminus \{1\}$ and $xyz=1$, prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$.
negative_passages.docid: 23048
negative_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: 52357
negative_passages.text: If $a,b \in G$, show that $ab^{n}a^{-1}=(aba^{-1})^{n}$.
negative_passages.docid: 114198
negative_passages.text: Showing that $y$ is irreducible in $\mathbb R[x,y]/(x^2+y^2-1)$.
negative_passages.docid: 31933
negative_passages.text: $x^2 = 1$ in the ring $\mathbb{Z}/n \mathbb{Z}$