Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE JACOBI SUMS OVER GALOIS RINGS AND ITS ABSOLUTE VALUES

  • Received : 2019.03.12
  • Accepted : 2020.01.28
  • Published : 2020.05.01
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Abstract

The Galois ring R of characteristic pn having pmn elements is a finite extension of the ring of integers modulo pn, where p is a prime number and n, m are positive integers. In this paper, we develop the concepts of Jacobi sums over R and under the assumption that the generating additive character of R is trivial on maximal ideal of R, we obtain the basic relationship between Gauss sums and Jacobi sums, which allows us to determine the absolute value of the Jacobi sums.

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References

  1. B. C. Berndt, R. J. Evans, and K. S. Williams, Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1998.
  2. S. Fan and W. Han, Character sums over Galois rings and primitive polynomials over finite fields, Finite Fields Appl. 10 (2004), no. 1, 36-52. https://doi.org/10.1016/S1071-5797(03)00041-8
  3. K. F. Ireland and M. I. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, 84, Springer-Verlag, New York, 1982.
  4. H. Ishibashi, The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4, Graphs Combin. 12 (1996), no. 1, 39-54. https://doi.org/10.1007/BF01858443
  5. Y. H. Jang and S. P. Jun, The Gauss sums over Galois rings and its absolute values, Korean J. Math. 26 (2018), no. 3, 519-535. https://doi.org/10.11568/kjm.2018.26.3.519
  6. J. Li, S. Zhu, and K. Feng, The Gauss sums and Jacobi sums over Galois ring GR($p^2$, r), Sci. China Math. 56 (2013), no. 7, 1457-1465. https://doi.org/10.1007/s11425-013-4629-6
  7. R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, London, 1997.
  8. B. R. McDonald, Finite Rings with Identity, Marcel Dekker, Inc., New York, 1974.
  9. Y. Oh and H. J. Oh, Gauss sums over Galois rings of characteristic 4, Kangweon-Kyungki Math. Jour. 9 (2001), no. 1, 1-7.
  10. Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. https://doi.org/10.1142/5350
  11. J. Wang, On the Jacobi sums modulo $P^n$, J. Number Theory 39 (1991), no. 1, 50-64. https://doi.org/10.1016/0022-314X(91)90033-8
  12. J. Wang, On the Jacobi sums for finite commutative rings with identity, J. Number Theory 48 (1994), no. 3, 284-290. https://doi.org/10.1006/jnth.1994.1068
  13. M. Yamada, Distance-regular digraphs of girth 4 over an extension ring of Z/4Z, Graphs Combin. 6 (1990), no. 4, 381-394. https://doi.org/10.1007/BF01787706