Abstract

L. Introduction. Ko will denote a real algebraic number field and K a real normal algebraic extension of Ko. If a = 1, U2, a3, . . ., a, are the elements of the Galois group G of K/Ko then for a E K, NmKIK0a= I ai, and an integer whose norm is one is a relative unit of K/Ko. When K is regarded as a subset of a real n-dimensional vector space over Ko, the relative units are contained in a hypersurface having n asymptotic hyperplanes. Hasse [2] has shown that in some real cyclic fields over the rationals there exists a unit with the property that n 1 of its conjugates and 1 generate the full group of units of K. In this paper we consider, for K cyclic over Ko, the multiplicative group generated by n -1 conjugates of an element in K which is not in Ko if n is an odd prime, and has conjugates whose squares are multiplicatively independent (see Definition 2) if n is not prime. It is shown here that these groups lie close to the asymptotic hyperplanes; more precisely, any infinite set of elements of each group contains a subset whose points at infinity converge to the infinite part of one of these asymptotic hyperplanes. Furthermore, the group generated by a single element of K whose square is in no subfield containing Ko, K not necessarily cyclic over Ko, lies close to the n lines which are the intersections of n 1 of the asymptotic hyperplanes. We obtain as corollaries some results concerning the finiteness of the number of elements of these groups belonging to algebraic varieties. In particular we show that any linear subvariety can contain only a finite number of powers of an element of K whose square is in no subfield containing Ko. Moreover, we obtain restrictive conditions on an algebraic variety that it contain infinitely many such points. We also show that no line can contain an infinite number of elements belonging to the group generated by n -1 conjugates of an element of the type being considered. The solutions to these above mentioned diophantine problems are obtained by algebraic methods and do not rely on the Thue-Siegel-Roth theorems. Some of the results in this paper were obtained earlier in [3]. There one of the proofs employed the Gelfond-Schneider theorem. This has recently been generalized by A. Baker [1], which provides a simplification of the proof and an extension of the result. ACKNOWLEDGMENT. The author wishes to thankProfessor Leon Ehrenpreis of New York University for his advice and encouragement in the preparation of this paper.

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