Variations on a Theme of Kronecker
- 1 June 1978
- journal article
- Published by Canadian Mathematical Society in Canadian Mathematical Bulletin
- Vol. 21 (2) , 129-133
- https://doi.org/10.4153/cmb-1978-023-x
Abstract
In 1857, Kronecker [10] showed that if θ1,…, θn are the roots of the polynomial P(z)= zn|cn-1+ … + cn, where c1, …, cn are integers with cn≠0, and if |θ1| ≤ 1, …, |θ1| ≤1, then θ1, …, θn are roots of unity. The proof is short and ingenious: Consider the polynomials Pm(z) whose roots are The condition on the size of the roots and the fact that the ci are integers implies that there can only be a finite number of different Pm. Thus two distinct powers of each root must coincide and this means that each root is a root of unity.Keywords
This publication has 10 references indexed in Scilit:
- Algebraic integers whose conjugates lie near the unit circleBulletin de la Société Mathématiques de France, 1978
- Small Salem numbersDuke Mathematical Journal, 1977
- On the Product of the Conjugates outside the unit circle of an Algebraic IntegerBulletin of the London Mathematical Society, 1971
- Algebraic integers near the unit circleActa Arithmetica, 1971
- On sets of algebraic integers whose remaining conjugates lie in the unit circleTransactions of the American Mathematical Society, 1962
- Power series with integral coefficientsDuke Mathematical Journal, 1945
- Algebraic integers whose conjugates lie in the unit circleDuke Mathematical Journal, 1944
- A remarkable class of algebraic integers. Proof of a conjecture of VijayaraghavanDuke Mathematical Journal, 1944
- Factorization of Certain Cyclotomic FunctionsAnnals of Mathematics, 1933
- Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten.Journal für die reine und angewandte Mathematik (Crelles Journal), 1857