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Normal integral bases and tameness conditions for Kummer extensions

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Del Corso I., Rossi L.

Acta Arithmetica

2013

tom 160

nr 1

1-23

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.

Ograniczanie wyników

1 Acta Arithmetica

1 Del Corso I.

1 Rossi L.

1 2013

Wyniki wyszukiwania

Normal integral bases and tameness conditions for Kummer extensions