query: $A=\frac{\mathbb{C}[X,Y]}{(X^2+Y^2-1)}$ is a PID.
positive_passages.docid: 153069
positive_passages.text: Proof for Unique Factorization Domain
negative_passages.docid: 139453
negative_passages.text: Prove that $\frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz$
negative_passages.docid: 65901
negative_passages.text: Solution verification to complex analysis problem, prove $f$ is a polynomial.
negative_passages.docid: 94869
negative_passages.text: Show that $ \int_{k}^{2} \frac{1}{f(y)} =1$
negative_passages.docid: 1859
negative_passages.text: Proving $\arctan(\sqrt{x^2+1}-x)=\frac{\pi}{4}-\frac{1}{2}\arctan(x)$
negative_passages.docid: 135577
negative_passages.text: Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
negative_passages.docid: 62755
negative_passages.text: The simplest way of proving that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}| = c$
negative_passages.docid: 117816
negative_passages.text: Is $\frac{x^2+x}{x+1}$ a polynomial?
negative_passages.docid: 145882
negative_passages.text: How prove this $\frac{a}{(bc-a^2)^2}+\frac{b}{(ca-b^2)^2}+\frac{c}{(ab-c^2)^2}=0$
negative_passages.docid: 116630
negative_passages.text: Antiderivative $1/z$ on $\mathbb C$
negative_passages.docid: 156646
negative_passages.text: If $x$ is rational then $\,x+\frac{1}x \,$ is an integer $\iff x = \pm 1$
negative_passages.docid: 129394
negative_passages.text: understanding $\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$
negative_passages.docid: 28332
negative_passages.text: Proof of $\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}$
negative_passages.docid: 117922
negative_passages.text: Is $\frac{a+b}{c+d}0$?
negative_passages.docid: 28987
negative_passages.text: One-dimensional [Noetherian] UFD is a PID
negative_passages.docid: 91786
negative_passages.text: Proving and disproving $\exists b,c\in \mathbb R$ such that $f(x)=\frac a2x^2+bx+c $
negative_passages.docid: 114418
negative_passages.text: A deduction from $x^n-1=(x^2-1)\prod_{k=1}^{(n-2)/2}[x^2-2x\cos{\frac{2k\pi}{n}}+1]$