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Acta Arithmetica

2013 | 160 | 1 | 1-23

Normal integral bases and tameness conditions for Kummer extensions

EN

Abstrakty

EN
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 $ℚ(ζ_m,\sqrt[m]{a_1},...,\sqrt[m]{a_n})/ℚ(ζ_m)$ 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.

1-23

wydano
2013

Twórcy

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