On small discriminants of number fields of degree 8 and 9

Battistoni, Francesco

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 489-501.

Nous classifions tous les corps de nombres de signature $(4,2),$ $(6,1),$ $(1,4)$ et $(3,3)$ et discriminant inférieur à une certaine borne spécifique. Ceci achève la recherche du discriminant minimal pour les corps de degré $8$ et contribue à l’étude du cas de degré $9.$ On rappelle les outils théoriques et les étapes algorithmiques sur lesquels repose notre méthode, on se concentre ensuite sur les aspects nouveaux qui proviennent de la nouvelle implémentation de ce processus dans le système de calcul formel PARI/GP ; enfin, on fait quelques remarques sur nos résultats finals, parmi lesquels mentionnons l’existence d’un corps de nombres de signature $(3,3)$ et d’un petit discriminant, inconnu jusqu’à présent.

We classify all the number fields with signature $(4,2),$ $(6,1),$ $(1,4)$ and $(3,3)$ having discriminant lower than a specific upper bound. This completes the search for minimum discriminants for fields of degree 8 and continues it in the degree 9 case. We recall the theoretical tools and the algorithmic steps upon which our procedure is based, then we focus on the novelties due to a new implementation of this process on the computer algebra system PARI/GP; finally, we make some remarks about the final results, among which the existence of a number field with signature $(3,3)$ and small discriminant which was not previously known.

Classification : 11R21, 11R29, 11Y40
Mots clés : Number fields, classification for small discriminant.

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