Normal integral bases and tameness conditions for Kummer extensions

• Autorzy: Ilaria Del Corso , Lorenzo Paolo Rossi
• Języki publikacji: 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.

Kategorie tematyczne

• 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers
• 11S15: Ramification and extension theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 160
• Numer: 1
• Strony: 1-23

Daty

wydano

2013

Twórcy

autor

Ilaria Del Corso

• Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo, 5, 56127 Pisa, Italy

autor

Lorenzo Paolo Rossi

• Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy

Identyfikatory

DOI

10.4064/aa160-1-1