An effective method to study the hopf-galois module structure of certain extensions of fields

Autores: Daniel Gil Muñoz
Directores de la Tesis: Anna Rio (dir. tes.) , Teresa Crespo Vicente (codir. tes.)
Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2021
Idioma: español
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Resumen
- We develop a method to compute a basis of the associated order in a Hopf Galois structure H of the ring of integers O_L of an extension of number or p-adic fields L/K. We state and prove a necessary and sufficient condition for a given element in O_L to be a free generator of O_L as module over its associated order in H. Whenever it exists, one can use such a free generator and a basis of this associated order to build a basis which can be seen as an analog of the normal integral basis in the Galois case. We use this method to determine the associated order and the existence of those normal integral bases generators for different classes of extensions of fields, such as Galois extensions of degrees 2, 3 and 4, and separable degree p extensions of Q_p with normal closure having Galois group isomorphic to the dihedral group D_p of 2p elements. We shall use the theory of induced Hopf Galois structures to study the same problem for the normal closure itself, i.e. a dihedral degree 2p extension of Q_p. We give complete answers for the cases p=3 and p=5.