## Πολυώνυμα με ειδικές ιδιότητες πάνω από πεπερασμένα σώματα

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PhD thesis (EN)

2015 (EN)
Polynomials with special properties over finite fields
Πολυώνυμα με ειδικές ιδιότητες πάνω από πεπερασμένα σώματα

Καπετανάκης, Γεώργιος Ν

Kontogeorgis, Aristides
Chlouveraki, Maria
Garefalakis, Theodoulos
Dais, Dimitrios
Tzanakis, Nikos
Poulakis, Dimitrios

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. (EN)

Τύπος Εργασίας--Διδακτορικές διατριβές
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English

2015-07-15

Σχολή/Τμήμα--Σχολή Θετικών και Τεχνολογικών Επιστημών--Τμήμα Μαθηματικών--Διδακτορικές διατριβές

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