On applications of finite fields to the Euler function

U. M. Pachev, A. A. Tokbaeva

Kabardino-Balkarian State University, 173 Chernyshevsky St., Nalchik 360004, Russia

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.

Key words: finite field, normed irreducible polynomial, polynomial order, $n$-cyclotomic polynomial, Euler function.

UDC: 511.17, 512.624

MSC: 11T55

Received: 28.04.2025

DOI: 10.46698/m2155-1449-8044-d