On the linear independence in algebraic number fields (Q1320082)

In a recent work of \textit{P. Bundschuh} and the author [Monatsh. Math. 117, 17-32 (1994; see the review above)] obtained a criterion of linear independence over \(\mathbb{Q}\) by generalizing the earlier linear independence considerations of \textit{Yu. V. Nesterenko} [Vestn. Mosk. Univ., Ser. I 1985, No. 1, 46-49 (1985; Zbl 0572.10027)]. In the present paper the author introduces a further development of the method for the consideration of criteria of linear independence over arbitrary algebraic number fields. The results are given also in the quantitative form and as an application an explicit quantitative version of the Lindemann- Weierstrass theorem is proved, see also a paper of \textit{M. Ably} [Acta Arith. 67, No. 1, 29-45 (1994)], where the Lindemann-Weierstrass theorem is considered by another method.