We extend the Euler totient function to the rings of algebraic integers of quadratic extensions of the rational numbers, which are unique factorization domains.

Euler function, quadratic field.

Received: May 4, 2021; Accepted: June 24, 2021; Published: August 21, 2021

How to cite this article: Javier Diaz-Vargas, Carlos Jacob Rubio-Barrios and Horacio Tapia-Recillas, The Euler Totient function on quadratic fields, JP Journal of Algebra, Number Theory and Applications 52(1) (2021), 17-94. DOI: 10.17654/NT052010017

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] L. E. Dickson, A generalization of Fermat’s theorem, Ann. Math. (2) 1(1/4) (1899-1900), 31-36.
[2] C. Y. Chao, Generalizations of theorems of Wilson, Fermat and Euler, J. Number Theory 15(1) (1982), 95-114.
[3] James T. Cross, The Euler Φ-function in the Gaussian integers, Amer. Math. Monthly 90(8) (1983), 518-528.
[4] J. Díaz Vargas and H. Tapia Recillas, La functión Φ de Euler en los Enteros de Eisenstein-Jacobi, Reportes de Investigación, Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, 1989.
[5] F. S. Gillespie, A generalization of Fermat’s little theorem, Fibonacci Quart. 27(2) (1989), 109-115.
[6] M. Lassák and S. Porubsky, Fermat-Euler theorem in algebraic number fields, J. Number Theory 60(2) (1996), 254-290.
[7] A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1997.
[8] E. H. Moore, A two-fold generalization of Fermat’s theorem, Bull. Amer. Math. Soc. 2(7) (1896), 189-199.
[9] I. Niven and L. J. Warren, A generalization of Fermat’s theorem, Proc. Amer. Math. Soc. 8(2) (1957), 306-313.
[10] I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, 5th ed., John Wiley & Sons Inc., 1991.
[11] R. T. Harger and R. M. Smith, A generalization of Fermat’s little theorem, Internat. J. Math. Ed. Sci. Tech. 31(3) (2000), 476-477. Doi:
10.1080/00207390050032351.
[12] B. Schneier, Applied Cryptography, John Wiley & Sons Inc. 2nd ed., 1966.
[13] W. Sierpinski, Elementary Theory of Numbers, Warsaw, 1964.
[14] D. R. Stinson, Cryptography Theory and Practice, CRC Press, 1995.
[15] A. V. Zarelua, On matrix analogues of Fermat’s little theorem, Math. Notes 79(5-6) (2006), 783-796.