MATHEMATICS OF COMPUTATION Volume 75, Number 255, July 2006, Pages 1507–1518 S 0025-5718(06)01899-0 Article electronically published on April 3, 2006 DEPENDENCY OF UNITS IN NUMBER FIELDS CLAUS FIEKER AND MICHAEL E. POHST Abstract. We develop a method for validating the indepencence of units in algebraic number fields. In case that a given system of units has a dependency, we compute a certificate for this. 1. Introduction A key problem in computational number theory is to decide whether a given system of units {ε 1 ,...,ε l } is (multiplicatively) independent, i.e., to decide if for some z =(z 1 ,...,z l ) ∈ Z l , z = 0, we have l  i=1 ε z i i =1. This problem occurs naturally during the computation of class groups or unit groups and is therefore important for most applications. There are known algorithmic solutions for this problem (e.g., the use of MLLL [8] or a real-gcd [3, Algorithm 6.5.7]), but they lack reliability in the sense that they utilize real arithmetic and do not provide rigorous error analysis. In our new approach, we use a different numerical method to check for depen- dencies (with an error analysis) and then use any of the above methods to find a dependency. 2. Notation Throughout this section and the subsequent ones F denotes an algebraic number field of degree d over the rational numbers Q. We assume that it is generated by a root ρ of a monic irreducible polynomial f (t)= t d + a 1 t d−1 + ··· + a d ∈ Z[t]. Over the complex numbers C the polynomial f (t) splits into a product of linear factors f (t)= d  j=1 (t − ρ (j) ), Received by the editor July 21, 2004. 2000 Mathematics Subject Classification. Primary 11Y16, 11-04. This article was written while the second author visited the Computational Algebra Group at the University of Sydney in October, 2003. c 2006 American Mathematical Society 1507 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use