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Groupes, corps et extensions de Polya : une question de capitulation

Abstract : In this thesis, we focus on the set $Int\left(\mathcal O _K \right)$ of integer-valued polynomials over $\mathcal{O}_K$, the ring of integers of a number field $K$. According to G. Pólya, a basis $\left(f_{n}\right)_{n\in \mathbb{N}}$ of the $\mathcal O _K$-module $Int\left(\mathcal O _K \right)$ is said to be regular if for each $n \in \mathbb{N}$, $\deg(f_{n})=n$. A field $K$ such that $Int\left(\mathcal O _K \right)$ has a regular basis is said to be a Pólya field and the Pólya group of number field $K$ is a subgroup of the class group of $K$ which can be considered as a measure of the obstruction for a field being a Pólya field. We study the Pólya group of a compositum $L= K_1 K_2$ of two galoisian extensions $K_1 /\mathbb{Q}$ and $K_2 /\mathbb{Q}$ and we link it to the behaviour of the ramification of primes in $K_1 /\mathbb{Q}$ and $K_2 /\mathbb{Q}$. We apply these results to number fields with small degree in order to enlarge the well known family of quadratic Pólya fields. Furthermore, a field $K$ is a Pólya field if the products of all maximal ideals of $\mathcal{O}_K$ with the same norm are principal. Analogously to the classical embedding problem, we can set the following problem : is every number field contained in a Pólya field? We give a positive answer to this question : for each number field $K$, the Hilbert class field $H_K$ of $K$ is a Pólya field. We know also that every ideal of $\mathcal{O}_K$ becomes principal in $\mathcal{O}_{H_K}$. This leads us to introduce the notion of Pólya extension : it is a field $L$ containing $K$ such that the Pólya group of $K$ becomes trivial by extension of ideals, it is also a field $L$ such that the $\mathcal O _L$-module generated by $Int\left(\mathcal O _K \right)$ has a regular basis. Consequently, $H_K$ is a Pólya extension of $K$ in the general case. Moreover, when $K$ is abelian, capitulation of ambigeous ideals of $K$ proves that the genus field of $K$ is a Pólya extension. This leads us to consider minimality and unicity questions for Pólya fields and Pólya extensions.
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Submitted on : Friday, July 29, 2011 - 12:29:08 PM
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Amandine Leriche. Groupes, corps et extensions de Polya : une question de capitulation. Mathématiques [math]. Université de Picardie Jules Verne, 2010. Français. ⟨tel-00612597⟩

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