query: Let $\alpha , \beta \in \mathbb{Z}[\sqrt{d}]$. Prove that $\alpha$ is a unit $\iff N(\alpha) = 1$.
positive_passages.docid: 92583
positive_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: 156766
negative_passages.text: Prove that $\sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$
negative_passages.docid: 145752
negative_passages.text: Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
negative_passages.docid: 24466
negative_passages.text: Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$
negative_passages.docid: 52966
negative_passages.text: Rings, ideals and units-proof
negative_passages.docid: 145509
negative_passages.text: Prove the roots of $p(z)+z^n\bar{p}(\frac{1}{z})$ lie on the unit circle
negative_passages.docid: 133528
negative_passages.text: $\lvert K(\alpha_1,\dots,\alpha_n) : K \rvert$ is a divisor of $n!$
negative_passages.docid: 153204
negative_passages.text: Euclidean domain $\mathbb{Z}[\sqrt{d}]$
negative_passages.docid: 56818
negative_passages.text: Prove that if $A$ is positive, then $\exists\;\alpha>0$ such that $\langle Ax, x \rangle\geq \alpha|x|^2$ for $x\in\Bbb{R}^n$
negative_passages.docid: 81775
negative_passages.text: Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$
negative_passages.docid: 86807
negative_passages.text: The units of $\mathbb Z[\sqrt{2}]$
negative_passages.docid: 88425
negative_passages.text: Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$
negative_passages.docid: 50934
negative_passages.text: Let $x_n \in[0,1)$ s.t $x_n - \sqrt{n}\in \mathbb{Z}$. Prove that $X=\{x_n | n\in \mathbb{N}\}$ is dense in $[0,1]$.
negative_passages.docid: 148550
negative_passages.text: Units in quotient ring of $\mathbb Z[X]$
negative_passages.docid: 90332
negative_passages.text: Prove that that $U(n)$ is an abelian group.
negative_passages.docid: 42677
negative_passages.text: Proving $(\alpha + \beta)^{p^n} = \alpha^{p^n} + \beta^{p^n}$ in a finite field
negative_passages.docid: 79679
negative_passages.text: Let $\alpha, \beta \in \mathbb{C}$, such that $\alpha + \beta$, and $\alpha\beta$ are algebraic. Show that $\alpha$ and $\beta$ are algebraic.