On Applications of Finite Fields to the Euler Function

		ISSN 1683-3414 (Print) • ISSN 1814-0807 (Online)

		Log in

Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/m2155-1449-8044-d On Applications of Finite Fields to the Euler Function Pachev, U. M. , Tokbaeva, A. A. Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 3. Abstract: The manuscript is devoted to applications of finite fields to the Euler function from number theory. Using the concept of a normalized irreducible polynomial of a given degree over a finite field \(F_q\), we obtain an analogue of the well known Gauss relation \(\sum_{d\|n}{\varphi(d) =}n\). Here \(\varphi(k)\) is the Euler arithmetic function such that its value is equal to the number of integers \(1,2,\ldots,k\) relatively prime to \(k\). In order to formulate and prove an analogue of this relation we use concepts and preliminary results from the polynomial theory over a finite field \(F_{q}\) of \(q\) elements. In particular, we apply the concepts of a normalized irreducible polynomial of one variable over the field \(F_{q}\) and \(n\)-circle polynomial \(Q_{n}(x)\) over any field of nonzero characteristic. In addition, we use the concept of the order of a polynomial \(f(x) \in F_{q}[x]\) such that if the polynomial \(f(x)\) divides \(x^{e} - 1\) in the ring \(F_{q}[x]\), then the minimal natural number \(e\) is the order of the polynomial \(f(x)\). The proof of the main new results is based on the explicit formula for the \(n\)-circle polynomial \(Q_n(x)\) and on the auxiliary result for the number of normalized irreducible polynomials \(f(x)\in F_{q}[x]\) degree \(m\) and given order \(e\). We obtain the formula for \(N_{q}(n)\) of normalized irreducible polynomials degree \(n\) and an analogue of the Gauss relation for the Euler function. Keywords: finite field, normed irreducible polynomial, polynomial order, \(n\)-cyclotomic polynomial, Euler function Language: Russian Download the full text Download JATS XML For citation: Pachev, U. M. and Tokbaeva, A. A.On Applications of Finite Fields to the Euler Function, Vladikavkaz Math. J., 2025, vol. 27, no. 3, pp. 120-126 (in Russian). DOI 10.46698/m2155-1449-8044-d + References 1. Gauss, K. F. Works on Number Theory, Moscow, Publishing House the USSR Academy of Sciences, 1959, 987 p. (in Russian). 2. Galois, E. From the Theory of Numbers. Works, Moscow, Leningrad, ONTI, 1936, pp. 35-47 (in Russian). 3. Pachev, U. M. Selected Chapters of Number Theory, Nalchik, "M. and V. Kotlyarov", 2016, 186 p. (in Russian). 4. Mineev, M. P. and Chubarikov, V. N. Lectures on Arithmetic Issues of Cryptography, Moscow, 2014, 224 p. (in Russian). 5. Stepanov, S. A. Congruences, Publishing house "Knowledge", Moscow, 1975 (in Russian). 6. Lidl, R. and Niederreiter, H. {Finite Fields. Vol. 1, Cambridge University Press, 1983. 7. Lenskoy, D. N. On the Arithmetic of Polynomials Over a Finite Field, Volzhsky Mat. Sb., Kuibishev. Works of Mathematical Departaments of Pedagogical Institutes of the Volga Region, issue 4, 1966, pp. 155-159 (in Russian). 8. Fredman, M. L. Congruence Formulas Obtained by Counting Irreducible, Pacific Journal of Mathematics, 1970, vol. 35, no. 3, pp. 613-624. DOI: 10.2140/pjm.1970.35.613. 9. Stepanov, S. A. On the Number of Irreducible Polynomials in $F_{q[x]$ of Special Form, Russian Mathematical Surveys, 1985, vol. 40, no. 4, pp. 219-220. DOI: 10.1070/RM1985v040n04ABEH003656. 10. Carlitz, L. The Arithmetic of Polynomials in a Galois Field, Proceedings of the National Academy of Sciences, 1931, vol. 17, no. 2, pp. 120-122. DOI: 10.1073/pnas.17.2.120. 11. Willams, K. S. Polynomials with Irreducible Factors of Specified Degree, Canadian Mathematical Bulletin, 1969, vol. 12 , no. 2, pp. 221-223. DOI: 10.4153/CMB-1969-026-2. 12. Pachev, U. M., Kodzokov, A. Kh. and Mashezova, A. M. On Estimates on the Number of Normalized Irreducible Polynomials of a Given Degree Over a Finite Field, Algebra and Dynamical System. Abstracts of Reports of the International Conference Dedicated to the 110-th Anniversary of the Birth of S. N. Chernikov, Nalchik, 2022, pp. 100-101. 13. Carlitz, L. Some Theorems on Irreducible Polynomials Over a Finite Field, Journal fur die reine und angewandte Mathematik, 1967, vol. 227, pp. 212-220. 14. Levine, J. and Brawley, J. V. Involutory Commutants with Some Applications to Algebraic Cryptography, Journal fur die Reine und Angewandte Mathematik, 1966, vol. 224, pp. 20-43. 15. Levine, J. and Brawley, J. V. Some Cryptographic Applications of Permutation Polynomials, Cryptologia, 1977, vol. 1, pp. 76-92. DOI: 10.1080/0161-117791832814. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


	\| Home \| Editorial board \| Publication ethics \| Peer review guidelines \| Latest issue \| All issues \| Rules for authors \| Online submission system's guidelines \| Submit manuscript \|
	© 1999-2026 Южный математический институт