Revista de la
Unión Matemática Argentina
Additive and multiplicative relations with algebraic conjugates
Artūras Dubickas and Paulius Virbalas

Volume 69, no. 1 (2026), pp. 143–154    

Published online (final version): December 19, 2025

https://doi.org/10.33044/revuma.5106

Download PDF

Abstract

We prove that every nontrivial additive relation between algebraic conjugates of degree $d$ over $\mathbb{Q}$ has a corresponding multiplicative relation. The proof is constructive. The reverse statement is not true. These findings supplement the research of Smyth, Dixon, Girstmair and others. In addition, following a result of Kitaoka we show that all additive relations between four distinct algebraic conjugates of degree 4 over $\mathbb{Q}$ can be described as $\mathbb{Z}$-linear combinations of several basic nontrivial relations. On the other hand, we prove that an analogous result no longer holds for algebraic conjugates of degree 6 over $\mathbb{Q}$.

References

  1. E. Artin, Galois theory, second ed., Dover, Mineola, NY, 1998.  MR  Zbl
  2. G. Baron, M. Drmota, and M. Skałba, Polynomial relations between polynomial roots, J. Algebra 177 no. 3 (1995), 827–846.  DOI  MR  Zbl
  3. J. D. Dixon, Polynomials with nontrivial relations between their roots, Acta Arith. 82 no. 3 (1997), 293–302.  DOI  MR  Zbl
  4. M. Drmota and M. Skałba, On multiplicative and linear independence of polynomial roots, in Contributions to general algebra 7 (Vienna, 1990), Hölder-Pichler-Tempsky, Vienna, 1991, pp. 127–135.  MR  Zbl
  5. M. Drmota and M. Skałba, Relations between polynomial roots, Acta Arith. 71 no. 1 (1995), 65–77.  DOI  MR  Zbl
  6. P. Drungilas and A. Dubickas, On degrees of three algebraic numbers with zero sum or unit product, Colloq. Math. 143 no. 2 (2016), 159–167.  DOI  MR  Zbl
  7. A. Dubickas, On the degree of a linear form in conjugates of an algebraic number, Illinois J. Math. 46 no. 2 (2002), 571–585.  MR  Zbl Available at http://projecteuclid.org/euclid.ijm/1258136212.
  8. A. Dubickas, Additive relations with conjugate algebraic numbers, Acta Arith. 107 no. 1 (2003), 35–43.  DOI  MR  Zbl
  9. A. Dubickas, Multiplicative relations with conjugate algebraic numbers, Ukraïn. Mat. Zh. 59 no. 7 (2007), 890–900, and in Ukrainian Math. J. 59 no. 7 (2007), 984–995.  DOI  MR  Zbl
  10. A. Dubickas and J. Jankauskas, Simple linear relations between conjugate algebraic numbers of low degree, J. Ramanujan Math. Soc. 30 no. 2 (2015), 219–235.  MR  Zbl
  11. A. Dubickas and C. J. Smyth, Polynomials with three distinct zeros summing to zero (problem 11123), Amer. Math. Monthly 113 no. 10 (2006), 941–942, solution by R. Stong.  DOI
  12. K. Girstmair, Linear dependence of zeros of polynomials and construction of primitive elements, Manuscripta Math. 39 no. 1 (1982), 81–97.  DOI  MR  Zbl
  13. K. Girstmair, Linear relations between roots of polynomials, Acta Arith. 89 no. 1 (1999), 53–96.  DOI  MR  Zbl
  14. K. Girstmair, The Galois relation $x_1=x_2+x_3$ and Fermat over finite fields, Acta Arith. 124 no. 4 (2006), 357–370.  DOI  MR  Zbl
  15. K. Girstmair, The Galois relation $x_1=x_2+x_3$ for finite simple groups, Acta Arith. 127 no. 3 (2007), 301–303.  DOI  MR  Zbl
  16. K. Girstmair, The Galois relations $x_1=x_2+x_3$ and $x_1=x_2x_3$ for certain solvable groups, Ann. Sci. Math. Québec 32 no. 2 (2008), 171–174.  MR  Zbl
  17. W. Hardt and J. Yin, Linear relations among Galois conjugates over $\mathbb{F}_q(t)$, Res. Number Theory 8 no. 2 (2022), Paper No. 34.  DOI  MR  Zbl
  18. Y. Kitaoka, Notes on the distribution of roots modulo a prime of a polynomial, Unif. Distrib. Theory 12 no. 2 (2017), 91–117.  MR  Zbl
  19. V. A. Kurbatov, Galois extensions of prime degree and their primitive elements, Izv. Vysš. Učebn. Zaved. Matematika no. 1(176) (1977), 61–66, English transl.: Soviet Math. (Iz. VUZ) 21 no. 1 (1977), 49–53.  MR  Zbl
  20. F. Lalande, Relations linéaires entre les racines d'un polynôme et anneaux de Schur, Ann. Sci. Math. Québec 27 no. 2 (2003), 169–175.  MR  Zbl
  21. F. Lalande, La relation linéaire $a=b+c+\dots+t$ entre les racines d'un polynôme, J. Théor. Nombres Bordeaux 19 no. 2 (2007), 473–484.  DOI  MR  Zbl
  22. F. Lalande, À propos de la relation galoisienne $x_1=x_2+x_3$, J. Théor. Nombres Bordeaux 22 no. 3 (2010), 661–673.  DOI  MR  Zbl
  23. C. J. Smyth, Conjugate algebraic numbers on conics, Acta Arith. 40 no. 4 (1982), 333–346.  DOI  MR  Zbl
  24. C. J. Smyth, Additive and multiplicative relations connecting conjugate algebraic numbers, J. Number Theory 23 no. 2 (1986), 243–254.  DOI  MR  Zbl
  25. P. Virbalas, Linear relations between three conjugate algebraic numbers of low degree, J. Korean Math. Soc. 62 no. 2 (2025), 253–284.  DOI  MR  Zbl
  26. T. Zheng, Characterizing triviality of the exponent lattice of a polynomial through Galois and Galois-like groups, in Computer algebra in scientific computing, Lecture Notes in Comput. Sci. 12291, Springer, Cham, 2020, pp. 621–641.  DOI  MR  Zbl
  27. T. Zheng, A fast algorithm for computing multiplicative relations between the roots of a generic polynomial, J. Symbolic Comput. 104 (2021), 381–401.  DOI  MR  Zbl