query: Show that $k[x,x^{-1}]$ is a PID
positive_passages.docid: 24224
positive_passages.text: How to prove the ring of Laurent polynomials over a field is a principal ideal domain?
negative_passages.docid: 27629
negative_passages.text: How to prove that $S^{2n+1}/S^1$ is homeomorphic to $\mathbb CP^n$ under a given identification
negative_passages.docid: 96592
negative_passages.text: Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc.
negative_passages.docid: 2907
negative_passages.text: How do I show that $x$ is a root of $p(x)$ in $F[x]/\langle p(x)\rangle$?
negative_passages.docid: 10540
negative_passages.text: Prove $\vdash \neg(\square F\land p)$ in $KD$
negative_passages.docid: 63104
negative_passages.text: Dual cone of $K = \{(x,t) \mid \| \boldsymbol{x} \|_1 \le t \}$
negative_passages.docid: 101371
negative_passages.text: show that $f^{(3)}(c) \ge 3$ for $c\in(-1,1)$
negative_passages.docid: 111129
negative_passages.text: Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.
negative_passages.docid: 7903
negative_passages.text: Prove that $B[x] \cap B[x^{-1}]$ is integral over $B$
negative_passages.docid: 44453
negative_passages.text: Prove that the normalisation of $A=k[X,Y]/(Y^2-X^2-X^3)$ is $k[t]$ where $t=Y/X$ (Reid, Exercise 4.5)
negative_passages.docid: 149783
negative_passages.text: $\mathbb Z\times\mathbb Z$ is principal but is not a PID
negative_passages.docid: 127365
negative_passages.text: Question on showing a bijection between $\pi_1(X,x_0)$ and $[S^1, X]$ when X is path connected.
negative_passages.docid: 48492
negative_passages.text: KL divergence of q(z|x) || p(z|x)
negative_passages.docid: 158062
negative_passages.text: Sum of free submodules of a module over a PID
negative_passages.docid: 154001
negative_passages.text: $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$
negative_passages.docid: 46211
negative_passages.text: $(K[x,y,z]/(x^2+y^3+z^7))_{(x,y,z)}$ is a UFD.
negative_passages.docid: 151805
negative_passages.text: Prove that $\tau (x_1 x_2 ... x_k) \tau^{-1} = (\tau(x_1) \tau(x_2) ... \tau (x_k))$