Timeline for Does there exist some $p(x) \in \mathbb{Q}[x]$, deg$(p) > 1$, which maps $\mathbb{Q}$ onto itself surjectively?

10 events

when toggle format	what		by	license	comment
Sep 19, 2021 at 15:25	comment	added	Yaakov Baruch		Stronger question answered here: mathoverflow.net/questions/186768/images-of-polynomials
Sep 19, 2021 at 2:51	answer	added	Joseph Van Name		timeline score: 8
Sep 18, 2021 at 22:48	comment	added	alpoge		(…because wlog by scaling $f\in \Z[x]$, and then when $p$ doesn’t divide the leading coefficient $c_0$ of $f(x) =: c_0 x^d + \cdots + c_d\in \Z[x]$ the $p$-adic valuation of ($(a,b) = 1$) $f(a/b) = b^{-d} (c_0 a^d + c_1 a^{d-1} b + \cdots + c_d b^d) = b^{-d} (c_0 a^d + (\in b\Z))$ is either $\geq 0$ (when $(p,b)=1$) or a multiple of $d$ (when $p\vert b$, whence $(p,a) = 1$), so it can’t be $-1$.)
Sep 18, 2021 at 18:06	answer	added	Joe Silverman		timeline score: 17
Sep 18, 2021 at 6:45	comment	added	Wlod AA		Perhaps $\ \mathbb Q\setminus f(\mathbb Q)\ $ is dense in $\ \mathbb Q$.
Sep 18, 2021 at 5:26	comment	added	alpoge		Nope cause you won’t hit $1/p$ for $p$ large (lemme know if I’m being stupid!).
Sep 18, 2021 at 4:44	vote	accept	Bma
Sep 18, 2021 at 4:42	answer	added	John Doyle		timeline score: 32
Sep 18, 2021 at 4:24	history	edited	Bma	CC BY-SA 4.0	added 108 characters in body
Sep 18, 2021 at 4:11	history	asked	Bma	CC BY-SA 4.0