Polynomials with special properties over finite fields
Πολυώνυμα με ειδικές ιδιότητες πάνω από πεπερασμένα σώματα
Καπετανάκης, Γεώργιος Ν
over finite fields. In Chapter 2 some background material is presented. We present
some basic concepts of characters of finite abelian groups and we prove some basic
results. Next, we focus on Dirichlet characters and on the characters of the additive
and the multiplicative groups of a finite field. We conclude this chapter with an expression
of the characteristic function of generators of cyclic R-modules, where R is a
Euclidean domain, known as Vinogradov’s formula.
In Chapter 3, we consider a special case of the Hansen-Mullen conjecture. In
particular, we consider the existence of self-reciprocal monic irreducible polynomials
of degree 2n over Fq, where q is odd, with some coefficient prescribed. First, we
use Carlitz’s characterization of self-reciprocal polynomials over odd finite fields and,
with the help of Dirichlet characters, we prove asymptotic conditions for the existence
of polynomials with the desired properties. As a conclusion, we restrict ourselves
to the first n=2 (hence also to the last n=2) coefficients, where our results are more
efficient, and completely solve the resulting problem.
In Chapter 4 we extend the primitive normal basis theorem and its strong version.
Namely, we consider the existence of polynomials whose roots are simultaneously
primitive, produce a normal basis and some given Möbius transformation of those
roots also produce a normal basis. First, we characterize elements with the desired
properties and with the help of characters, we end up with some sufficient conditions,
which we furtherly relax using sieving techniques. In the end, we prove our
desired results, with roughly the same exceptions as the ones appearing in the strong
primitive normal basis theorem.
In Chapter 5, we work in the same pattern as in Chapter 4, only here we demand
that the Möbius transformation of the roots of the polynomial is also primitive. We
roughly follow the same steps and prove that there exists a polynomial over a finite
field such that its roots are simultaneously primitive and produce a normal basis and
some given Möbius tranformation of its roots also possess both properties, given that
the cardinality of the field and the degree of the polynomial are large enough.