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ON FINITE GALOIS STABLE SUBGROUPS OF GL_n IN SOME RELATIVE EXTENSIONS OF NUMBER FIELDS

H.-J. BARTELS

Fakultät für Mathematik und Informatik, Universität Mannheim, Seminargebäude A5, D-68131 Mannheim, Germany

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and

D. A. MALININ

Fakultät für Mathematik und Informatik, Universität Mannheim, Seminargebäude A5, D-68131 Mannheim, Germany

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https://doi.org/10.1142/S0219498809003400Cited by:2 (Source: Crossref)

Abstract

Let K/ℚ be a finite Galois extension with maximal order and Galois group Γ. For finite Γ-stable subgroups it is known [4], that they are generated by matrices with coefficients in , K_ab the maximal abelian subextension of K over ℚ. This note gives a contribution to the corresponding question in the case of a relative Galois extension K/R, where R is a finite extension of the rationals ℚ. It turns out, that in this relative situation the answer to the corresponding question depends heavily on the arithmetic of the number field R, more precisely on the ramification behavior of primes in K/R. Due to the possibility of unramified extensions of R for certain number fields R there exist examples of Galois stable linear groups which are not fixed elementwise by the commutator subgroup of Gal(K/R).

Keywords:

AMSC: 20C10, 11R33, 11S23, 11R29

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