On Applications of Finite Fields to the Euler Function

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 Fq, we obtain an analogue of the well known Gauss relation Pd|n '(d) =n. Here '(k) is the Euler arithmetic function such that its value is equal to the number of integers 1, 2, . . . , 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 Fq of q elements. In particular, we apply the concepts of a normalized irreducible polynomial of one variable over the field Fq and n-circle polynomial Qn(x) over any field of nonzero characteristic. In addition, we use the concept of the order of a polynomial f(x) ∈ Fq[x] such that if the polynomial f(x) divides xe−1 in the ring Fq[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 Qn(x) and on the auxiliary result for the number of normalized irreducible polynomials f(x) ∈ Fq[x] degree m and given order e. We obtain the formula for Nq(n) of normalized irreducible polynomials degree n and an analogue of the Gauss relation for the Euler function.