A criterion for the linear independence of polylogarithms over a number field (Algebraic Number Theory and Related Topics 2014)

"Algebraic Number Theory and Related Topics 2014". December 1～5, 2014. edited by Takeshi Tsuji, Hiroki Takahashi and Yuichiro Hoshi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.Let Lis(z) be the s-th polylogarithmic function. Let α ∈ Q, 0 <｜2｜< 1. In this article, we give a criterion for the linear independence of the s+1 numbers Li1(α), Li2(α), …, Lis(α) and 1 over the number field Q(α). The new part is that we prove the linear independence of such polylogarithms over a number field of arbitrary degree. We also show examples and a linear independence measure