Abstract
Suppose k is a local field that is an extension of the field of p -adic numbers of degree n and does not contain a primitive p -th root of 1, and suppose K/k is a cyclic p-extension with Galois group G. The group E of principal units of K is a multiplicatively written module over the group ring Λ=ℤp[G], where ℤp is the ring of p-adic integers. It was shown by Borevich (Ref. Zh. Mat., 1965, 3A256) that the Λ-module E has a system of n+1 generators, of which n−1 are free and two are connected by certain relations. In the present paper these Λ-generators are constructed explicitly and their arithmetical characteristics indicated.
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Literature cited
Z. I. Borevich, “The multiplicative group of a regular local field with cyclic operator group,” Izv. Akad. Nauk SSSR, Ser. Mat.,28, No. 3, 707–712 (1964).
J.-P. Serre, Corps Locaux, Hermann, Paris (1962).
H. Hasse, Zahlentheorie, Akademie Verlag, Berlin (1963).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 16–23, 1977.
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Arutyunyan, L.Z. Generators of the group of principal units of a cyclic p-extension of a regular local field. J Math Sci 20, 2509–2515 (1982). https://doi.org/10.1007/BF01681467
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DOI: https://doi.org/10.1007/BF01681467