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Variations on a Theme of Kronecker

Published online by Cambridge University Press:  20 November 2018

David W. Boyd
David W. Boyd*
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, Canada V6T 1W5
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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.

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 2 , 01 June 1978, pp. 129 - 133
Copyright
Copyright © Canadian Mathematical Society 1978

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