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ON SUBFIELDS OF A FIELD GENERATED BY TWO CONJUGATE ALGEBRAIC NUMBERS

Published online by Cambridge University Press:  27 May 2004

Paulius Drungilas  and
Artūras Dubickas
Paulius Drungilas
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius 2600, Lithuania ([email protected]; [email protected])
Artūras Dubickas
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius 2600, Lithuania ([email protected]; [email protected])
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Abstract

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Let $k$ be a field, and let $\alpha$ and $\alpha'$ be two algebraic numbers conjugate over $k$. We prove a result which implies that if $L\subset k(\alpha,\alpha')$ is an abelian or Hamiltonian extension of $k$, then $[L:k]\leq[k(\alpha):k]$. This is related to a certain question concerning the degree of an algebraic number and the degree of a quotient of its two conjugates provided that the quotient is a root of unity, which was raised (and answered) earlier by Cantor. Moreover, we introduce a new notion of the non-torsion power of an algebraic number and prove that a monic polynomial in $X$—irreducible over a real field and having $m$ roots of equal modulus, at least one of which is real—is a polynomial in $X^m$.

AMS 2000 Mathematics subject classification: Primary 11R04; 11R20; 11R32; 12F10

Keywords

fieldsubfieldalgebraic numbersabelian and Hamiltonian extensionsroots of unity
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 47 , Issue 1 , February 2004 , pp. 119 - 123
Copyright
Copyright © Edinburgh Mathematical Society 2004