On Mahler's Classification of Formal Power Series Over a Finite Field

Gülcan Kekeç

doi:10.1515/ms-2022-0017

Abstract

Abstract Let K be a finite field, K(x) be the field of rational functions in x over K and K be the field of formal power series over K. We show that under certain conditions integral combinations with algebraic formal power series coefficients of a U 1-number in K are Um -numbers in K , where m is the degree of the algebraic extension of K(x), determined by these algebraic formal power series coefficients.

Highlights

Let K be a finite field with q elements
We denote the ring of polynomials in x with coefficients from K by K[x], the quotient field of K[x] by K(x) and the degree of a non-zero polynomial f (x) in K[x] by deg(f )
We denote the unique extension of | · | to the field K by the same notation | · |

Summary

The field of formal power series over a finite field

Let K be a finite field with q elements. We denote the ring of polynomials in x with coefficients from K by K[x], the quotient field of K[x] by K(x) and the degree of a non-zero polynomial f (x) in K[x] by deg(f ). The field K of formal power series over K is the completion of K(x) with respect to | · |. A transcendental) formal power series if it is algebraic Let P (y) = f0 + f1y + · · · + fnyn be a non-zero polynomial in y over K[x]. Let α be an algebraic formal power series and P (y) be its minimal polynomial over K[x].

Mahler’s classification of formal power series over a finite field

Construction of our main result

Auxiliary results