query: F is a field $\Rightarrow$ $F\left[ x\right]$ is a PIR
positive_passages.docid: 133561
positive_passages.text: show that if $K$ is a field then $K[x]$ is principal
negative_passages.docid: 136770
negative_passages.text: Any field that contains the integers contains the rationals as a subfield (Analysis proof)
negative_passages.docid: 144445
negative_passages.text: Functions $f:\mathbb{N}\rightarrow \mathbb{Z}$ such that $\left(m-n\right) \mid \left(f(m)-f(n)\right)$
negative_passages.docid: 104757
negative_passages.text: If $f \circ g$ is surjective, then f is surjective
negative_passages.docid: 138911
negative_passages.text: Is $\mathbb R^2$ a field?
negative_passages.docid: 80961
negative_passages.text: Prove that a commutative ring $F$ is a field iff $F[x]$ is a Principal Ideal Domain
negative_passages.docid: 61162
negative_passages.text: Show that $\left(\int_0^1 f(x)\,dx\right)^2\leq2\int_0^1x\, f(x)^2\,dx $
negative_passages.docid: 89271
negative_passages.text: If $F$ is a field, then $F[x]$ is a principal ideal domain proof question
negative_passages.docid: 98540
negative_passages.text: If $F[x]$ is a principal domain does $F$ have to be necessarily a field?
negative_passages.docid: 48291
negative_passages.text: Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $
negative_passages.docid: 89637
negative_passages.text: Proof that in a field $x=0$ is equivalent to $x=-x$
negative_passages.docid: 33586
negative_passages.text: Prove the field of fractions of $F[[x]]$ is the ring $F((x))$ of formal Laurent series.
negative_passages.docid: 3604
negative_passages.text: If F is a field, then $F[x,y]$ is a Principal Ideal Domain?
negative_passages.docid: 136855
negative_passages.text: Proof that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a PID
negative_passages.docid: 19838
negative_passages.text: Prove that in general the set of the right zero divisors is different from the set of left zero divisors in a ring R
negative_passages.docid: 87103
negative_passages.text: Quick question about ideals of field $F$ in $F[x]$ are principal.
negative_passages.docid: 134840
negative_passages.text: Generators and relations for $\text{PSL}(2, \mathbb{F}_q)$