Enumeration of a special class of irreducible polynomials in characteristic 2
$A$-polynomials were introduced by Meyn and play an important role in the iterative construction of high degree self-reciprocal irreducible polynomials over the field $\mathbb F_2$, 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 finite fields of characteristic 2. 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, with an isolated exception, there exist $A$-polynomials of every degree.