## Abstract

Let p be a prime number, Q_{p} the field of p-adic numbers, Q_{p} a fixed algebraic closure of Q_{p}, and C_{p} the completion of Q_{p}. For elements T∈C_{p} which satisfy a certain diophantine condition (*) we construct a power series F(T, Z) with coefficients in Q_{p} 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 C_{p} except on the set of conjugates of 1/T. The main result (Theorem 7.2) asserts that if {T_{α}}_{α} is a family of elements of C_{p} which are non-conjugate, transcendental over Q_{p}, and satisfy condition (*) then the functions {F(T_{α}, Z)}_{α} are algebraically independent over C_{p}(Z). In particular, if T is an element of C_{p} which satisfies condition (*), then F(T, Z) is transcendental over C_{p}(Z) if and only if T is transcendental over Q_{p}. 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 C_{p}.

Original language | English (US) |
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Pages (from-to) | 13-48 |

Number of pages | 36 |

Journal | Journal of Number Theory |

Volume | 88 |

Issue number | 1 |

DOIs | |

State | Published - May 2001 |

Externally published | Yes |

## ASJC Scopus subject areas

- Algebra and Number Theory

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