Roots of Unity as Quotients of Two Conjugate Algebraic Numbers
نویسندگان
چکیده
Let α be an algebraic number of degree d > 2 over Q. Suppose for some pairwise coprime positive integers n1, . . . , nr we have deg(αj ) < d for j = 1, . . . , r, where deg(α) = d for each positive proper divisor n of nj . We prove that then φ(n1 . . . nr) 6 d, where φ stands for the Euler totient function. In particular, if nj = pj , j = 1, . . . , r, are any r distinct primes satisfying deg(αj ) < d, then the inequality (p1 − 1) · · · (pr − 1) 6 d holds, and therefore r ≪ log d/ log log d for d > 3. This bound on r improves that of Dobrowolski r 6 log d/ log 2 proved in 1979 and is best possible.
منابع مشابه
Simultaneous Approximation by Conjugate Algebraic Numbers in Fields of Transcendence Degree One
We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers ξ1, ..., ξt by conjugate algebraic numbers of bounded degree over Q, provided that the given transcendental numbers ξ1, ..., ξt generate over Q a field of transcendence degree one. We provide sharper estimates for example when ξ1, ..., ξt form an arithmetic progression with non-ze...
متن کاملBost–connes Systems, Categorification, Quantum Statistical Mechanics, and Weil Numbers
In this article we develop a broad generalization of the classical Bost-Connes system, where roots of unit are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unit, Weil restriction, algebraic numbers, Weil numbers, CM fields, germs, completion of Weil numbers, etc. Making use of the Tannakian formalism, we categorify th...
متن کاملIrreducibility, Homoclinic Points and Adjoint Actions of Algebraic Z–Actions of Rank One
In this paper we consider Z-actions, d ≥ 1, by automorphisms of compact connected abelian groups which contain at least one expansive automorphism (such actions are called algebraic Z-actions of expansive rank one). If α is such a Z-action on an infinite compact connected abelian group X, then every expansive element α of this action has a dense group ∆αn(X) of homoclinic points. For different ...
متن کاملKronecker-weber plus Epsilon
We say that a group is almost abelian if every commutator is central and squares to the identity. Now let G be the Galois group of the algebraic closure of the field Q of rational numbers in the field C of complex numbers. Let Gab+ be the quotient of G universal for continuous homomorphisms to almost abelian profinite groups, and let Qab+ /Q be the corresponding Galois extension. We prove that ...
متن کاملThe Hecke Algebras of Type B and D and Subfactors
We define a nontrivial homomorphism from the Hecke algebra of type B onto a reduced algebra of the Hecke algebra of type A at roots of unity. We use this homomorphism to describe semisimple quotients of the Hecke algebra of type B at roots of unity. Using these quotients we determine subfactors obtained from the inclusion of Hecke algebra of type A into Hecke algebras of type B. We also study i...
متن کامل