# Enumeration of a special class of irreducible polynomials in characteristic 2

## TL;DR

This paper studies A-polynomials over characteristic 2 fields, relating them to elliptic function field extensions, and provides an explicit counting formula demonstrating their existence across degrees.

## Contribution

It extends the construction of A-polynomials to arbitrary even finite fields and derives an explicit enumeration formula using elliptic function field theory.

## Key findings

- Explicit counting formula for A-polynomials in characteristic 2
- Existence of A-polynomials for all degrees except one
- Connection between A-polynomials and inert places in elliptic function fields

## Abstract

A-polynomials were introduced by Meyn and play an important role in the iterative construction of high degree self-reciprocal irreducible polynomials over the field F_2, since they constitute the starting point of the iteration. The exact number of A-polynomials of each degree was given by Niederreiter. Kyuregyan extended the construction of Meyn to arbitrary even finite fields. We relate the A-polynomials in this more general setting to inert places in a certain extension of elliptic function fields and obtain an explicit counting formula for their number. In particular, we are able to show that, except for an isolated exception, there exist A-polynomials of every degree.