Algebraic independence of p-adic numbers

摘要

We prove lower bounds for the transcendence degree of fields generated by values of the p-adic exponential function. In particular, we estimate the transcendence degree of the field Q(e(alpha 1) ,..., e(alpha d)), where alpha(1) ,..., alpha(d) are algebraic (over the field of rational numbers) p-adic numbers that form a basis of a finite extension of Q.