An Extension of the Theorem on Primitive Divisors in Algebraic Number Fields

The theorem about primitive divisors in algebraic number fields is generalized in the following manner. Let A, B be algebraic integers, (A, B) = 1 , AB ^ 0, A/B not a root of unity, and Çk a primitive root of unity of order k . For all sufficiently large n , the number A" C,kB" has a prime ideal factor that does not divide Am £,'kBm for arbitrary m < n and j < k . The analogue of Zsigmondy's theorem in algebraic number fields [3] asserts the following. If A, B are algebraic integers, (A, B) = I, AB ^ 0, and A/B of degree d is not a root of unity, there exists a constant no(d) such that for n > no(d), A" Bn has a prime ideal factor that does not divide Am Bm for m < n . This theorem will be extended as follows: Theorem. Let K be an algebraic number field, A, B integers of K, (A, B) = 1, AB ± 0, A/B of degree d not a root of unity, and Ck a primitive kth root of unity in K. For every e > 0 there exists a constant c(d, e) such that if n > c(d, e)(l -r-logÄ:)1+£, there exists a prime ideal of K thatdivides An-C,kBn , but does not divide Am ÇJkBm for m < n and arbitrary j. The above theorem implies the finiteness of the number of solutions of generalized cyclotomic equations considered by Browkin [1, p. 236]. The proof will follow closely the proof given in [3]. Let Q(A/B) = Kq , 5 = §, where a, ß e Ko, a, ß are integers, and (a, ß) = D. Let S and So be the set of all isomorphic injections of Ko(Ck) and Ko , respectively, in the complex field, and set w(a/ß) = log II max{|aCT|, \ßa\} log Aö, oeSo where N denotes the absolute norm in K0 . Here, w(a/ß) is the logarithm of the Mahler measure of a/ß and so it is independent of the choice of a, ß in Lemma 1. If \a\ = \ß\, but a/ß is not a root of unity, then log\att-Ckßn\ = nlog\ß\ + 0(d + w(a/ß))lo$kn, Received by the editor July 27, 1992. 1991 Mathematics Subject Classification. Primary 11R04. ©1993 American Mathematical Society 0025-5718/93 $1.00+ $.25 per page