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No / Finite quantum systems in which the position and momentum take values in the Galois field GF(p), are studied. Ideas from the subject of field extension are transferred in the context of quantum mechanics. The Frobenius automorphisms in Galois fields lead naturally to the "Frobenius formalism" in a quantum context. The Hilbert space splits into "Frobenius subspaces" which are labeled with the irreducible polynomials associated with the yp¿y. The Frobenius maps transform unitarily the states of a Galois quantum system and leave fixed all states in some of its Galois subsystems (where the position and momentum take values in subfields of GF(p)). An analytic representation of these systems in the -sheeted complex plane shows deeper links between Galois theory and Riemann surfaces. ©2006 American Institute of Physics

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: See the full text for the abstract / AFRIKAANSE OPSOMMING: Sien die volteks vir die opsomming

We study the Hopf-Galois module structure of algebraic integers in some finite extensions of $ p $-adic fields and number fields which are at most tamely ramified. We show that if $ L/K $ is a finite unramified extension of $ p $-adic fields which is Hopf-Galois for some Hopf algebra $ H $ then the ring of algebraic integers $ \OL $ is a free module of rank one over the associated order $ \AH $. If $ H $ is a commutative Hopf algebra, we show that this conclusion remains valid in finite ramified extensions of $ p $-adic fields if $ p $ does not divide the degree of the extension. We prove analogous results for finite abelian Galois extensions of number fields, in particular showing that if $ L/K $ is a finite abelian domestic extension which is Hopf-Galois for some commutative Hopf algebra $ H $ then $ \OL $ is locally free over $ \AH $. We study in greater detail tamely ramified Galois extensions of number fields with Galois group isomorphic to $ C_{p} \times C_{p} $, where $ p $ is a prime number. Byott has enumerated and described all the Hopf-Galois structures admitted by such an extension. We apply the results above to show that $ \OL $ is locally free over $ \AH $ in all of the Hopf-Galois structures, and derive necessary and sufficient conditions for $ \OL $ to be globally free over $ \AH $ in each of the Hopf-Galois structures. In the case $ p = 2 $ we consider the implications of taking $ K = \Q $. In the case that $ p $ is an odd prime we compare the structure of $ \OL $ as a module over $ \AH $ in the various Hopf-Galois structures.

Demonstramos um teorema da correspondência do tipo Galois para ações de álgebras de Hopf pontuais de dimensão finita em álgebras primas. A correspondência acontece entre subálgebras racionalmente completas e comódulo subálgebras. As subálgebras racionalmente completas são subálgebras da álgebra prima, enquanto os comódulo subálgebras são comódulo subálgebras do produto smash entre o centralizador da álgebra prima em sua álgebra de quocientes de Martindale simétrica e a álgebra de Hopf. / A Galois-type correspondence theorem for prime algebras acted upon by a finite dimensional pointed Hopf algebra is proved. The correspondence involves rationally complete subalgebras and comodule subalgebras. The rationally complete subalgebras are subalgebras of the prime algebra, while the comodule subalgebras are comodule subalgebras of the smash product between the centralizer of the prime algebra in its symmetric Martindale quotient algebra and the Hopf algebra.

Neste trabalho, apresentamos um estudo sobre a existência ou não de bases normais auto-duais para extensões galoisianas finitas de corpos, mostrando que toda extensão galoisiana finita de grau ímpar posui uma base normal auto-dual, enquanto que para extensões galoisianas de grau par, apresentamos algumas condições suficientes que garantem a não existência de bases normais auto-duais / In this work, we present a study about the existence or not of self-dual normal bases for finite galoisian extensions of fields, showing that all the odd degree finite galoisian extension has a self-dual normal base, whereas for even degree galoisian extensions, we present some sufficient conditions that assure the non-existence of self-dual normal bases

Soit g = 1 ou 2 et p > 3 un nombre premier. Pour le groupe symplectique GSp2g, les systèmes de valeurs propres de Hecke apparaissant dans les espaces de formes automorphes classiques, d’un niveau modéré fixé et de poids variable, sont interpolés p-adiquement par un espace rigide analytique, la vari´et´e de Hecke pour GSp2g. Un sous-domaine suffisamment petit de cette variété peut être décrit comme l’espace rigide analytique associé `a une algèbre profinie T. Une composante irréductible de T est d´efinie par un anneau profini I et un morphisme θ : T → I. Dans le cas résiduellement irréductible on peut associer `a θ une représentation ρθ : Gal(Q/Q) → GSp2g(I). On étudie l’image de ρθ quand θ décrit une composante de pente positive de T. Pour g = 1 il s’agit d’un travail en commun avec A. Lovita et J. Tilouine. On suppose que g = 1 o`u que g = 2 et θ est résiduellement de type cube sym2trique. On montre que Im ρθ est “grande” et que sa taille est li´ee aux “congruences fortuites” de θ avec les transferts de familles pour groupes de rang plus petit. Plus précisement, on agrandit un sous-anneau I0de I[1/p] en un anneau B et on définit une sous-algèbre de Lie G de gsp2g(B) associée `a Im ρθ. On prouve qu’il existe un idéal non-nul l de I0 tel que l · sp2g(B) ⊂ G. Pour g = 1 les facteurs premiers de l correspondent aux points CM de la famille θ. Pour g = 2 les facteurs premiers de l correspondent `a des congruences fortuites de θ avec des sous-familles de dimension 0 ou 1, obtenues par des transferts de type cube sym´etrique de points ou familles de la courbe de Hecke pour GL2. / Let g = 1 or 2 and p > 3 be a prime. For the symplectic group GSp2g the Hecke eigensystems appearing in the spaces of classical automorphic forms, of a fixed tame level and varying weight, are p-adically interpolated by a rigid analytic space, the GSp2g-eigenvariety. A sufficiently small subdomain of the eigenvariety can be described as the rigid analytic space associated with a profinite algebra T. An irreducible component of T is defined by a profinite ring I and a morphism θ : T → I. In the residually irreducible case we can attach to θ a representation ρθ : Gal(Q/Q) → GSp2g(I). We study the image of ρθ when θ describes a positive slope component of T. In the case g = 1 this is a joint work with A. Iovita and J. Tilouine. Suppose either that g = 1 or that g = 2 and θ is residually of symmetric cube type. We prove that Im ρθ is “big” and that its size is related to the “accidental congruences” of θ with the subfamilies that are obtained as lifts of families for groups of smaller rank. More precisely, we enlarge a subring I0 of I[1/p] to a ring B and we define a Lie subalgebra G of gsp2g(B) associated with Im ρθ. We prove that there exists a non-zero ideal l of I0 such that l · sp2g(B) ⊂ G. For g = 1 the prime factors of l correspond to the CM points of the family θ. Such points do not define congruences between θ and a CM family, so we call them accidental congruence points. For g = 2 the prime factors of l correspond to accidental congruences of θ with subfamilies of dimension 0 or 1 that are symmetric cube lifts of points or families of the GL2-eigencurve.

Demonstramos um teorema da correspondência do tipo Galois para ações de álgebras de Hopf pontuais de dimensão finita em álgebras primas. A correspondência acontece entre subálgebras racionalmente completas e comódulo subálgebras. As subálgebras racionalmente completas são subálgebras da álgebra prima, enquanto os comódulo subálgebras são comódulo subálgebras do produto smash entre o centralizador da álgebra prima em sua álgebra de quocientes de Martindale simétrica e a álgebra de Hopf. / A Galois-type correspondence theorem for prime algebras acted upon by a finite dimensional pointed Hopf algebra is proved. The correspondence involves rationally complete subalgebras and comodule subalgebras. The rationally complete subalgebras are subalgebras of the prime algebra, while the comodule subalgebras are comodule subalgebras of the smash product between the centralizer of the prime algebra in its symmetric Martindale quotient algebra and the Hopf algebra.