Algebraic Independence of Certain Liouville Numbers in Non-Archimedean Fields

The theory of Liouville numbers has played a fundamental role in transcendence theory since Liouville’s pioneering construction of explicit transcendental numbers. A classical result of Adams [‘On the algebraic independence of certain Liouville numbers’, J. Pure Appl. Algebra 13 (1) (1978), 41–47] established the algebraic independence of certain Liouville series associated with multiplicatively independent integer bases. We prove non-Archimedean analogues of Adam’s theorem in two distinct settings: the field of p -adic numbers and function fields. We establish algebraic independence results for families of Liouville-type series defined in these settings, extending Adams’ theorem from the classical real case to the non-Archimedean framework.