Construction of irreducible polynomials through rational transformations

Abstract

Let ${\mathbb{F}}_q$ be the finite field with $q$ elements, where $q$ is a power of a prime. We discuss recursive methods for constructing irreducible polynomials over ${\mathbb{F}}_q$ of high degree using rational transformations. In particular, given a divisor $D>2$ of $q+1$ and an irreducible polynomial $f\in {\mathbb{F}}_q[x]$ of degree $n$ such that $n$ is even or $D\not\equiv 2(\mathrm{mod}4)$, we show how to obtain from $f$ a sequence $\{f_i{\}}_{i\ge 0}$ of irreducible polynomials over ${\mathbb{F}}_q$ with $\mathrm{deg}(f_i)=n\cdot D^i$.

21 pages; comments are welcome!

Cited by in corpus (1)

• On iterations of rational functions over perfect fields