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Cyclotomic polynomials (in the parallel worlds of number theory)
(Stellenbosch : Stellenbosch University, 201112)ENGLISH ABSTRACT: It is well known that the ring of integers Z and the ring of polynomials A = Fr[T] over a finite field Fr have many properties in common. It is due to these properties that almost all the famous ... 
Drinfeld modules and their application to factor polynomials
(Stellenbosch : Stellenbosch University, 201212)ENGLISH ABSTRACT: Major works done in Function Field Arithmetic show a strong analogy between the ring of integers Z and the ring of polynomials over a nite eld Fq[T]. While an algorithm has been discovered to factor ... 
Elliptic curve cryptography
(Stellenbosch : Stellenbosch University, 201612)ENGLISH ABSTRACT : In this thesis we present a selection of DiffieHellman cryptosystems, which were classically formulated using the multiplicative group of a finite field, but which may be generalised to use other group ... 
Geometric actions of the absolute Galois group
(Stellenbosch : University of Stellenbosch, 200603)This thesis gives an introduction to some of the ideas originating from A. Grothendieck's 1984 manuscript Esquisse d'un programme. Most of these ideas are related to a new geometric approach to studying the absolute ... 
Geometry of Complex Polynomials: On Sendov's Conjecture
(Stellenbosch : Stellenbosch University, 201612)ENGLISH ABSTRACT : Sendov’s conjecture states that if all the zeroes of a complex polynomial P(z) of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of P(z). In a ... 
Riemann hypothesis for the zeta function of a function field over a finite field
(Stellenbosch : Stellenbosch University, 201312)ENGLISH ABSTRACT: See the full text for the abstract