The conjugate dimension of algebraic numbers
| dc.creator | Berry, Neil | |
| dc.creator | Dubickas, Arturas | |
| dc.creator | Elkies, Noam D. | |
| dc.creator | Poonen, Bjorn | |
| dc.creator | Smyth, Chris | |
| dc.date | 2003-08-07 | |
| dc.date | 2004-05-04 | |
| dc.date.accessioned | 2026-07-07T05:00:14Z | |
| dc.date.available | 2026-07-07T05:00:14Z | |
| dc.description | We find sharp upper and lower bounds for the degree of an algebraic number in terms of the $Q$-dimension of the space spanned by its conjugates. For all but seven nonnegative integers $n$ the largest degree of an algebraic number whose conjugates span a vector space of dimension $n$ is equal to $2^n n!$. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order of finite subgroups of $GL_n(Q)$; this result depends on the classification of finite simple groups. In particular, we construct an algebraic number of degree 1152 whose conjugates span a vector space of dimension only 4. We extend our results in two directions. We consider the problem when $Q$ is replaced by an arbitrary field, and prove some general results. In particular, we again obtain sharp bounds when the ground field is a finite field, or a cyclotomic extension of $Q$. Also, we look at a multiplicative version of the problem by considering the analogous rank problem for the multiplicative group generated by the conjugates of an algebraic number. | |
| dc.description | 17 pages | |
| dc.identifier | https://arxiv.org/abs/math/0308069 | |
| dc.identifier | http://arxiv.org/abs/math/0308069 | |
| dc.identifier | Quart. J. Math. 55 (2004), 237-252 | |
| dc.identifier | doi:10.1093/qmath/hah003 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/68271 | |
| dc.subject | Number Theory | |
| dc.subject | 11R06 | |
| dc.title | The conjugate dimension of algebraic numbers | |
| dc.type | text |