The conjugate dimension of algebraic numbers
Identifieur interne : 000D72 ( Main/Exploration ); précédent : 000D71; suivant : 000D73The conjugate dimension of algebraic numbers
Auteurs : Neil Berry [Royaume-Uni] ; Artu Ras Dubickas [Lituanie] ; Noam D. Elkies [États-Unis] ; Bjorn Poonen [États-Unis] ; Chris Smyth [Royaume-Uni]Source :
- The Quarterly Journal of Mathematics [ 0033-5606 ] ; 2004-09.
English descriptors
- KwdEn :
- Acts trivially, Algebraic, Algebraic number, Algebraic numbers, Auxiliary polynomial, Cambridge university press, Conjugate dimension, Conjugate rank, Dmax, Eld, Galois, Galois extension, Galois extensions, Galois group, Galois theory, Hilbert irreducibility, Hilbertian, Irreducible, Minimal polynomial, Monic polynomial, Multiplicative, Multiplicative group, Multiplicative subgroup, Nite, Nite galois extension, Nite group, Nite subgroup, Nite subgroups, Other hand, Permutation group, Regular representation, Roposition, Semidirect product, Separable closure, Square root, Subgroup, Subrepresentation, Transcendental extension, Upper bounds, Vector space.
- Teeft :
- Acts trivially, Algebraic, Algebraic number, Algebraic numbers, Auxiliary polynomial, Cambridge university press, Conjugate dimension, Conjugate rank, Dmax, Eld, Galois, Galois extension, Galois extensions, Galois group, Galois theory, Hilbert irreducibility, Hilbertian, Irreducible, Minimal polynomial, Monic polynomial, Multiplicative, Multiplicative group, Multiplicative subgroup, Nite, Nite galois extension, Nite group, Nite subgroup, Nite subgroups, Other hand, Permutation group, Regular representation, Roposition, Semidirect product, Separable closure, Square root, Subgroup, Subrepresentation, Transcendental extension, Upper bounds, Vector space.
Abstract
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 non‐negative integers n the largest degree of an algebraic number whose conjugates span a vector space of dimension n is equal to 2nn!. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order of finite subgroups of GLn(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 Q(ω ℓ) 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.
Url:
DOI: 10.1093/qmath/hah003
Affiliations:
- Lituanie, Royaume-Uni, États-Unis
- Californie, Massachusetts, Écosse
- Cambridge (Massachusetts), Édimbourg
- Université Harvard, Université d'Édimbourg
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For all but seven non‐negative integers n the largest degree of an algebraic number whose conjugates span a vector space of dimension n is equal to 2nn!. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order of finite subgroups of GLn(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 Q(ω ℓ) 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.</div></front></TEI><affiliations><list><country><li>Lituanie</li><li>Royaume-Uni</li><li>États-Unis</li></country><region><li>Californie</li><li>Massachusetts</li><li>Écosse</li></region><settlement><li>Cambridge (Massachusetts)</li><li>Édimbourg</li></settlement><orgName><li>Université Harvard</li><li>Université d'Édimbourg</li></orgName></list><tree><country name="Royaume-Uni"><region name="Écosse"><name sortKey="Berry, Neil" sort="Berry, Neil" uniqKey="Berry N" first="Neil" last="Berry">Neil Berry</name></region><name sortKey="Smyth, Chris" sort="Smyth, Chris" uniqKey="Smyth C" first="Chris" last="Smyth">Chris Smyth</name></country><country name="Lituanie"><noRegion><name sortKey="Dubickas, Artu Ras" sort="Dubickas, Artu Ras" uniqKey="Dubickas A" first="Artu Ras" last="Dubickas">Artu Ras Dubickas</name></noRegion></country><country name="États-Unis"><region name="Massachusetts"><name sortKey="Elkies, Noam D" sort="Elkies, Noam D" uniqKey="Elkies N" first="Noam D." last="Elkies">Noam D. Elkies</name></region><name sortKey="Poonen, Bjorn" sort="Poonen, Bjorn" uniqKey="Poonen B" first="Bjorn" last="Poonen">Bjorn Poonen</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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