JEDNOSTKA NAUKOWA KATEGORII A+

# Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

## Normal integral bases and tameness conditions for Kummer extensions

### Tom 160 / 2013

Acta Arithmetica 160 (2013), 1-23 MSC: 11R33, 11S15. DOI: 10.4064/aa160-1-1

#### Streszczenie

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.

#### Autorzy

• Ilaria Del CorsoDipartimento di Matematica
Università di Pisa
Largo B. Pontecorvo, 5
56127 Pisa, Italy
e-mail
• Lorenzo Paolo RossiScuola Normale Superiore
Piazza dei Cavalieri, 7
56126 Pisa, Italy
e-mail

## Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Odśwież obrazek