Trace on Cp

Victor Alexandru, Nicolae Popescu, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review


Let p be a prime number, Qp the field of p-adic numbers, Qp a fixed algebraic closure of Qp, and Cp the completion of Qp. For elements T∈Cp which satisfy a certain diophantine condition (*) we construct a power series F(T, Z) with coefficients in Qp and show that two elements T, U produce the same series F(T, Z)=F(U, Z) if and only if they are conjugate. We view the coefficient of Z in F(T, Z) as the trace of T. Further, we study F(T, Z) viewed as a rigid analytic function and prove that it is defined everywhere on Cp except on the set of conjugates of 1/T. The main result (Theorem 7.2) asserts that if {Tα}α is a family of elements of Cp which are non-conjugate, transcendental over Qp, and satisfy condition (*) then the functions {F(Tα, Z)}α are algebraically independent over Cp(Z). In particular, if T is an element of Cp which satisfies condition (*), then F(T, Z) is transcendental over Cp(Z) if and only if T is transcendental over Qp. In proving these results we develop some additional machinery, to be also used in a forthcoming paper which continues the study of orbits of elements in Cp.

Original languageEnglish (US)
Pages (from-to)13-48
Number of pages36
JournalJournal of Number Theory
Issue number1
StatePublished - May 2001
Externally publishedYes

ASJC Scopus subject areas

  • Algebra and Number Theory


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