cyclotomic polynomials and units in cyclotomic number fields
The author proves (theorem 1) that if P(x)\(\neq x\) is a monic irreducible polynomial with integer coefficients such that its resultant with infinitely many cyclotomic polynomials is \(+1\) or -1, then P(x) is a cyclotomic polynomial. From this he deduces a number of interesting corollaries: for example, if \(\alpha\neq 0\) is an algebraic integer such that for infinitely many n, 1- \(\alpha\) n is a unit in the ring of algebraic integers \(Z[\alpha]\), then \(\alpha\) is a root of unity. The proof uses a theorem of \textit{A. Baker} on linear forms in the logarithms of algebraic numbers [see Acta Arith. 21, 117-129 (1972; Zbl 0244.10031)]. Thus the finite number of exceptions in theorem 1 and its corollaries can be effectively determined for a given noncyclotomic P(x).
scientific article; zbMATH DE number 4031818