Polynomials with roots in ${\Bbb Q}_p$ for all $p$
| dc.creator | Sonn, Jack | |
| dc.date | 2006-12-18 | |
| dc.date | 2007-01-28 | |
| dc.date.accessioned | 2026-07-07T07:43:08Z | |
| dc.date.available | 2026-07-07T07:43:08Z | |
| dc.description | Let $f(x)$ be a monic polynomial in $\dZ[x]$ with no rational roots but with roots in $\dQ_p$ for all $p$, or equivalently, with roots mod $n$ for all $n$. It is known that $f(x)$ cannot be irreducible but can be a product of two or more irreducible polynomials, and that if $f(x)$ is a product of $m>1$ irreducible polynomials, then its Galois group must be a union of conjugates of $m$ proper subgroups. We prove that for any $m>1$, every finite solvable group which is a union of conjugates of $m$ proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with $m=2$) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric--i.e. regular-- extension of $\dQ(t)$. | |
| dc.description | 6 pages, revised to simplify a proof, improve a result, add a remark, and make some minor corrections | |
| dc.identifier | https://arxiv.org/abs/math/0612528 | |
| dc.identifier | http://arxiv.org/abs/math/0612528 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/122708 | |
| dc.subject | Number Theory | |
| dc.subject | 11R32; 12F12 | |
| dc.title | Polynomials with roots in ${\Bbb Q}_p$ for all $p$ | |
| dc.type | text |