Permutation-Type Formulas of Frobenius Map and Their Applications

Zhibo Chen1
1Department of Mathematics Penn State University, McKeesport PA 15132, U.S.A.

Abstract

Let \(\mathbb{F}_q = \text{GF}(q^n)\) denote the finite field of order \(q^n\), and let \(U_n = \cup_{i=1}^{n}(\mathbb{F}_i,)\). Explicit permutation-type formulas for the Frobenius map \(\varphi\) (defined by \((\varphi(x)) = x^q\)) on \(\mathbb{F}_n\) and on \(U_n\) are obtained by using the well-known number \(N_q\) (the number of monic irreducible polynomials of degree \(i\) in \(\mathbb{F}_1[x]\)). Some results in [1] and [2] can be obtained from these formulas. Moreover, some other results are also given by using Pólya’s counting theory.