Enumeration of a special class of irreducible polynomials in characteristic 2
Volume 194 / 2020
Acta Arithmetica 194 (2020), 51-57
MSC: Primary 11T55, 11R58; Secondary 14H05, 14Q05.
DOI: 10.4064/aa190116-21-5
Published online: 31 January 2020
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 $\mathbb F_2$, 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.
