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# Séminaire des Doctorants

Le 25 mai 2018
à 16:30
*Salle 2*
Francesco Battistoni (UniMi)
**Classification of number fields via discriminants**
The discriminant dK of a number field K is an integer which detects the rational prime numbers ramifying in K. A classical result by Hermite implies that for every M > 0 there are only finitely many number fields K with |dK| ≤ M. This started an attempt to classify all number fields which have discriminant less than a given bound.Meanwhile, it is well known, thanks to Minkowski, that, for every number field K of degree n, the minimal admissible value for |dK| increases with respect to n, and more precise lower bounds occur when one considers fields with a fixed number of real embeddings in C.In this seminar we will give a survey of the study of number fields via their discriminants, focusing on the analytic estimates for the lower bounds of |dK| produced by Odlyzko, Poitou and Serre and on the geometric-algorithmic methods by Hunter, Pohst and Martinet for the classification of fields with discriminant bounded from above. It will be shown how the combination of these methods allowed to get tables of number fields up to isomorphism.