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On the maximum modulus of integers in Kummer extensions

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Abstract

We study the extension of a result of Loxton [5] on representation of algebraic integers as sums of roots of unity to Kummer extensions.

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References

  1. A. Bérczes, A. Ostafe, I. Shparlinski and J. H. Silverman, Multiplicative dependence among iterated values of rational functions modulo finitely generated groups, Internat. Math. Res. Notices (to appear)

  2. J. W. S. Cassels, On a conjecture of R. M. Robinson about sums of roots of unity, J. Reine Angew. Math., 238 (1969) 112–131

  3. Dvornicich, R., Zannier, U.: Cyclotomic diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps). Duke Math. J. 139, 527–554 (2007)

    Article  MathSciNet  Google Scholar 

  4. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press (1969)

    MATH  Google Scholar 

  5. J. H. Loxton, On the maximum modulus of cyclotomic integers, Acta Arith., 22 69–85

  6. M. R. Murty and J. Esmonde, Problems in Algebraic Number Theory, 2nd ed., Graduate Texts in Mathematics, vol. 190, Springer-Verlag (New York, 2005)

  7. J. Neukirch, Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften, vol. 322, Springer-Verlag (Berlin, 1999)

  8. Ostafe, A.: On roots of unity in orbits of rational functions. Proc. Amer. Math. Soc. 145, 1927–1936 (2017)

    Article  MathSciNet  Google Scholar 

  9. A. Ostafe, M. Sha, I. E. Shparlinski and U. Zannier, On abelian multiplicatively dependent points on a curve in a torus, Q. J. Math., to appear

  10. A. Ostafe, M. Sha, I. E. Shparlinski and U. Zannier, Multiplicative independence among values of rational functions, and a generalisation of Northcott's theorem, Michigan Math. J., to appear

  11. S. Weintraub, Galois Theory, Springer (New York, 2006)

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  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    J. Mello

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  1. J. Mello
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Mello, J. On the maximum modulus of integers in Kummer extensions. Acta Math. Hungar. 164, 66–84 (2021). https://doi.org/10.1007/s10474-020-01117-3

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  • DOI: https://doi.org/10.1007/s10474-020-01117-3

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