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On the Product of the Conjugates outside the unit circle of an Algebraic Integer

On the Product of the Conjugates outside the unit circle of an Algebraic Integer ON THE PRODUCT OF THE CONJUGATES OUTSIDE THE UNIT CIRCLE OF AN ALGEBRAIC INTEGER C. J. SMYTH 1. Introduction Let a ( ^ 0, 1) be an algebraic integer with conjugates a = a a ,... , a over 1? 2 n the rationals. Let P(z) be the minimal polynomial of a, and Q(z) = z" P(z~ ). We prove the following result: THEOREM. Let 6 be the real root of9 -9-1 = 0. If then P(z) is a reciprocal polynomial (i.e. P(z) = Q(z)). The constant 0 ( = 1 • 32 ...) is clearly best possible. As immediate consequences of this theorem, we have alternative proofs of the following well-known results: COROLLARY 1. 0 is the smallest PV-number. The original proof is due to Siegel [1]. (A PK-number is an algebraic integer a with |a| > 1 and |<Xy| < 1 (j = 2, ..., «).) COROLLARY 2. The smallest complex PV -numbers have absolute value \/6 . Chamfy [2] first proved this result. In fact a must be either + i J6 or a non-real conjugate of ±9 ~ . (A complex PK-number a is a non-real algebraic integer with |oc| = |a| > 1 and http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On the Product of the Conjugates outside the unit circle of an Algebraic Integer

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/3.2.169
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Abstract

ON THE PRODUCT OF THE CONJUGATES OUTSIDE THE UNIT CIRCLE OF AN ALGEBRAIC INTEGER C. J. SMYTH 1. Introduction Let a ( ^ 0, 1) be an algebraic integer with conjugates a = a a ,... , a over 1? 2 n the rationals. Let P(z) be the minimal polynomial of a, and Q(z) = z" P(z~ ). We prove the following result: THEOREM. Let 6 be the real root of9 -9-1 = 0. If then P(z) is a reciprocal polynomial (i.e. P(z) = Q(z)). The constant 0 ( = 1 • 32 ...) is clearly best possible. As immediate consequences of this theorem, we have alternative proofs of the following well-known results: COROLLARY 1. 0 is the smallest PV-number. The original proof is due to Siegel [1]. (A PK-number is an algebraic integer a with |a| > 1 and |<Xy| < 1 (j = 2, ..., «).) COROLLARY 2. The smallest complex PV -numbers have absolute value \/6 . Chamfy [2] first proved this result. In fact a must be either + i J6 or a non-real conjugate of ±9 ~ . (A complex PK-number a is a non-real algebraic integer with |oc| = |a| > 1 and

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1971

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