Normal integral bases and tameness conditions for Kummer extensions

Ilaria Del Corso ¹   ; Lorenzo Paolo Rossi ²

¹ Dipartimento di Matematica Università di Pisa Largo B. Pontecorvo, 5 56127 Pisa, Italy
² Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa, Italy

Acta Arithmetica, Tome 160 (2013) no. 1, pp. 1-23

Résumé

We present a detailed analysis of some properties of a general tamely ramified Kummer extension of number fields $L/K$. Our main achievement is a criterion for the existence of a normal integral basis for a general Kummer extension, which generalizes the existing results. Our approach also allows us to explicitly describe the Steinitz class of $L/K$ and we get an easy criterion for this class to be trivial. In the second part of the paper we restrict to the particular case of tame Kummer extensions $\mathbb {Q}(\zeta _m,\sqrt [m]{a_1},\dots ,\sqrt [m]{a_n})/\mathbb {Q}(\zeta _m)$ with $a_i\in \mathbb {Z}$. We prove that these extensions always have trivial Steinitz classes. We also give sufficient conditions for the existence of a normal integral basis for such extensions and an example showing that such conditions are sharp in the general case. A detailed study of the ramification produces explicit necessary and sufficient conditions on the elements $a_i$ for the extension to be tame.

Détail

DOI : 10.4064/aa160-1-1

Keywords: present detailed analysis properties general tamely ramified kummer extension number fields main achievement criterion existence normal integral basis general kummer extension which generalizes existing results approach allows explicitly describe steinitz class get easy criterion class trivial second part paper restrict particular tame kummer extensions mathbb zeta sqrt dots sqrt mathbb zeta mathbb prove these extensions always have trivial steinitz classes sufficient conditions existence normal integral basis extensions example showing conditions sharp general detailed study ramification produces explicit necessary sufficient conditions elements extension tame