Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern

Abstract

Let K be an algebraic number field with non-zero α , β ∈ K . Siegel showed in 1929 that there are only finitely many units ε 1 , ε 2 in K which satisfy the unit equation αε 1 + βε 2 =1. In this article we present a new algorithm for solving unit equations which utilizes methods from the geometry of numbers. For the fist time unit equations in number fields up to unit rank 10 and with more than 100,000 solutions are solved. By applying our algorithm to index form equations we compute all power integral bases in the cyclotomic number fields up to degree 12 and in Q ( ζ 17 ), Q ( ζ 19 ), Q ( ζ 23 ).