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Adeles and class fields
Tížková, Bára ; Kala, Vítězslav (advisor) ; Gajović, Stevan (referee)
The first aim of the thesis is to study the ring of adèles and the group of idèles and work out their topology in detail. We explain the relation between the restricted product topology and other topologies which might seem natural on these objects. Further, we study their compactness properties. The second aim of the thesis is to summarize the main results of class field theory, both in the language of ideals and of idèles, and to provide examples illustrating new notions and concepts. Roughly speaking, class field theory describes all abelian extensions of a number field in terms of some "inner arithmetic" of the field. First, we demonstrate how the description works on two particular types of abelian extensions and then generalize the notions for any abelian extension. In order to present the general class field theory in a clear and straightforward manner, we unify the content of various sources and literature. The classical approach via ideals is more natural; however, some inconveniences arise when we have to take into account which primes ramify in the extension. This is handled by the idèlic approach described in the last chapter. 1
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