TY - JOUR
T1 - Trace on Cp
AU - Alexandru, Victor
AU - Popescu, Nicolae
AU - Zaharescu, Alexandru
N1 - Funding Information:
1This research was partially supported by the Deutscher Akademischer Austauschdienst.
PY - 2001/5
Y1 - 2001/5
N2 - 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.
AB - 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.
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U2 - 10.1006/jnth.2000.2610
DO - 10.1006/jnth.2000.2610
M3 - Article
AN - SCOPUS:0035343180
SN - 0022-314X
VL - 88
SP - 13
EP - 48
JO - Journal of Number Theory
JF - Journal of Number Theory
IS - 1
ER -