Timeline for Does there exist some $p(x) \in \mathbb{Q}[x]$, deg$(p) > 1$, which maps $\mathbb{Q}$ onto itself surjectively?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 7, 2021 at 7:42 | comment | added | abx | Oh, sure! Thank you. | |
| Oct 6, 2021 at 10:42 | comment | added | Joseph Van Name | Suppose that $q(x)=a_{n}x^{n}+\dots+a_{0}$. Let $q_{1}(x)=q(x)-a_{0}$. Then $q_{1}(x)=a_{n}x^{n}+\dots+a_{1}x$. Let $q_{2}(x)=q_{1}(a)/a_{n}$. Then $q_{2}(x)=x^{n}+b_{n-1}x^{n-1}+\dots+b_{1}x$. Now select a non-zero integer $b$ such that $b^{n-k}b_{k}$ is an integer for $1\leq k<n$. Then let $p(x)=x^{n}+b\cdot b_{n-1}x^{n-1}+\dots+b^{n-1}\cdot b_{1}x$. Then $p(x)=b^{n}q_{2}(x/b)$. | |
| Oct 6, 2021 at 7:50 | comment | added | abx | I don't see how replacing $q$ by $\alpha (q(\beta x)-\gamma )$ produces a monic polynomial with integer coefficients. | |
| Sep 19, 2021 at 7:13 | comment | added | Bma | Thanks for a great answer. This is the most elementary one yet. | |
| Sep 19, 2021 at 2:51 | history | answered | Joseph Van Name | CC BY-SA 4.0 |