Normal integral bases and tameness conditions for Kummer extensions

Ilaria Del Corso, and Lorenzo Paolo Rossi. "Normal integral bases and tameness conditions for Kummer extensions." Acta Arithmetica 160.1 (2013): 1-23. <http://eudml.org/doc/279720>.

@article{IlariaDelCorso2013,
abstract = {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 $ℚ(ζ_m,\@root m \of \{a_1\},...,\@root m \of \{a_n\})/ℚ(ζ_m)$ with $a_i ∈ ℤ$. 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.},
author = {Ilaria Del Corso, Lorenzo Paolo Rossi},
journal = {Acta Arithmetica},
keywords = {Kummer extensions; NIB; tame extensions},
language = {eng},
number = {1},
pages = {1-23},
title = {Normal integral bases and tameness conditions for Kummer extensions},
url = {http://eudml.org/doc/279720},
volume = {160},
year = {2013},
}

TY - JOUR
AU - Ilaria Del Corso
AU - Lorenzo Paolo Rossi
TI - Normal integral bases and tameness conditions for Kummer extensions
JO - Acta Arithmetica
PY - 2013
VL - 160
IS - 1
SP - 1
EP - 23
AB - 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 $ℚ(ζ_m,\@root m \of {a_1},...,\@root m \of {a_n})/ℚ(ζ_m)$ with $a_i ∈ ℤ$. 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.
LA - eng
KW - Kummer extensions; NIB; tame extensions
UR - http://eudml.org/doc/279720
ER -