Self-Dual Integral Normal Bases and Galois Module Structure

Let $N/F$ be an odd degree Galois extension of number fields with Galois group $G$ and rings of integers ${\mathfrak O}_N$ and {\mathfrak O}_F=\bo respectively. Let $\mathcal{A}$ be the unique fractional ${\mathfrak O}_N$-ideal with square equal to the inverse different of $N/F$. Erez has shown that $\mathcal{A}$ is a locally free ${\mathfrak O}[G]$-module if and only if $N/F$ is a so called weakly ramified extension. There have been a number of results regarding the freeness of $\mathcal{A}$ as a $\Z[G]$-module, however this question remains open. In this paper we prove that $\mathcal{A}$ is free as a $\Z[G]$-module assuming that $N/F$ is weakly ramified and under the hypothesis that for every prime $\wp$ of ${\mathfrak O}$ which ramifies wildly in $N/F$, the decomposition group is abelian, the ramification group is cyclic and $\wp$ is unramified in F/\Q. We make crucial use of a construction due to the first named author which uses Dwork's exponential power series to describe self-dual integral normal bases in Lubin-Tate extensions of local fields. This yields a new and striking relationship between the local norm-resolvent and Galois Gauss sum involved. Our results generalise work of the second named author concerning the case of base field \Q

Éléments explicites en théorie algébrique des nombres

This report consists in a synthesis of my research activities in algebraic number theory, between 2003 and 2013, on my own or with colleagues. The main goal is the study of the Galois module structure of modules associated to number field extensions, under various hypothesis, specifically about ramification. We also present results about other subjects which came into the way of the previous study: the construction of a certain type of Galois extensions of the field of rationals, the complexity of self-dual normal bases for multiplication in finite fields, and a bit of combinatorics. We stress the importance of an explicit knowledge of the objects under study.Ce mémoire présente une synthèse de mes travaux de recherche en théorie algébrique des nombres menés entre 2003 et 2013, seul ou en collaboration. Ils portent principalement sur l'étude de la structure galoisienne de modules associés à des extensions de corps de nombres, sous diverses hypothèses en particulier de ramification. Ils abordent aussi des thèmes rencontrés chemin faisant : construction d'un certain type d'extensions galoisiennes du corps des rationnels, complexité des bases normales auto-duales pour la multiplication dans les corps finis, un peu de combinatoire. Dans la présentation de tous ces travaux, l'accent est mis sur l'aspect explicite des objets étudiés

Vegetation patch effects on flow resistance at channel scale

International audienceThanks to a specific experimental design in a controlled channel, this paper aimed at quantifying how patches of four different ditches plant species affect integrated flow resistance parameters, the Manning coefficient. These plants, frequently encountered in the farmland ditches and irrigation channels of the south of France, were selected according to a large range of hydrophilic requirements, flexibility and branching complexity related to the plant blockage factor. Eight different spatial patches (regular, random, lateral or central patches) of each plant with crescent or similar plant densities were implanted at the bottom of a controlled channel where the water levels and water velocities were measured for three different discharges in steady and unsteady flow conditions. Resistance parameters (Manning parameters) were then estimated from the total head-loss, or from flow propagation velocity in the channel thanks to inversion of an hydrodynamic model. These experiments allow us to test the significance effect of channel vegetation patches and densities on flow resistance parameters at the reach scale

Construction of self-dual normal bases and their complexity

Recent work of Pickett has given a construction of self-dual normal bases for extensions of finite fields, whenever they exist. In this article we present these results in an explicit and constructive manner and apply them, through computer search, to identify the lowest complexity of self-dual normal bases for extensions of low degree. Comparisons to similar searches amongst normal bases show that the lowest complexity is often achieved from a self-dual normal basis

Dynamic simulation of the THAI heavy oil recovery process

Toe-to-Heel Air Injection (THAI) is a variant of conventional In-Situ Combustion (ISC) that uses a horizontal production well to recover mobilised partially upgraded heavy oil. It has a number of advantages over other heavy oil recovery techniques such as high recovery potential. However, existing models are unable to predict the effect of the most important operational parameters, such as fuel availability and produced oxygen concentration, which will give rise to unsafe designs. Therefore, we have developed a new model that accurately predicts dynamic conditions in the reservoir and also is easily scalable to investigate different field scenarios. The model used a three component direct conversion cracking kinetics scheme, which does not depend on the stoichiometry of the products and, thus, reduces the extent of uncertainty in the simulation results as the number of unknowns is reduced. The oil production rate and cumulative oil produced were well predicted, with the latter deviating from the experimental value by only 4%. The improved ability of the model to emulate real process dynamics meant it also accurately predicted when the oxygen was first produced, thereby enabling a more accurate assessment to be made of when it would be safe to shut-in the process, prior to oxygen breakthrough occurring. The increasing trend in produced oxygen concentration following a step change in the injected oxygen rate by 33 % was closely replicated by the model. The new simulations have now elucidated the mechanism of oxygen production during the later stages of the experiment. The model has allowed limits to be placed on the air injection rates that ensure stability of operation. Unlike previous models, the new simulations have provided better quantitative prediction of fuel laydown, which is a key phenomenon that determines whether, or not, successful operation of the THAI process can be achieved. The new model has also shown that, for completely stable operation, the combustion zone must be restricted to the upper portion of the sand pack, which can be achieved by using higher producer back pressure

Galois module structure in weakly ramified 3-extensions

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Une famille infinie d'extensions faiblement ramifiées

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Permuting the partitions of a prime

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