ON SUBFIELDS OF A FIELD GENERATED BY TWO CONJUGATE ALGEBRAIC NUMBERS

Author:

Drungilas Paulius,Dubickas Artūras

Abstract

AbstractLet $k$ 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

Publisher

Cambridge University Press (CUP)

Subject

General Mathematics

Cited by 7 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Cyclotomic Quotients of Two Conjugates of an Algebraic Number;Siberian Mathematical Journal;2021-05

2. Roots of unity as quotients of two conjugate algebraic numbers;Glasnik Matematicki;2017-11-13

3. Counting degenerate polynomials of fixed degree and bounded height;Monatshefte für Mathematik;2014-08-19

4. ROOTS OF UNITY AS QUOTIENTS OF TWO ROOTS OF A POLYNOMIAL;Journal of the Australian Mathematical Society;2012-04

5. On the discriminant of the power of an algebraic number;Studia Scientiarum Mathematicarum Hungarica;2007-03-01

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