Abstract view
The Generating Degree of $\C_p$


Published:20010301
Printed: Mar 2001
Victor Alexandru
Nicolae Popescu
Alexandru Zaharescu
Abstract
The generating degree $\gdeg (A)$ of a topological commutative ring
$A$ with $\Char A = 0$ is the cardinality of the smallest subset $M$
of $A$ for which the subring $\Z[M]$ is dense in $A$. For a prime
number $p$, $\C_p$ denotes the topological completion of an algebraic
closure of the field $\Q_p$ of $p$adic numbers. We prove that $\gdeg
(\C_p) = 1$, \ie, there exists $t$ in $\C_p$ such that $\Z[t]$ is
dense in $\C_p$. We also compute $\gdeg \bigl( A(U) \bigr)$ where
$A(U)$ is the ring of rigid analytic functions defined on a ball $U$
in $\C_p$. If $U$ is a closed ball then $\gdeg \bigl( A(U) \bigr) =
2$ while if $U$ is an open ball then $\gdeg \bigl( A(U) \bigr)$ is
infinite. We show more generally that $\gdeg \bigl( A(U) \bigr)$ is
finite for any {\it affinoid} $U$ in $\PP^1 (\C_p)$ and $\gdeg \bigl(
A(U) \bigr)$ is infinite for any {\it wide open} subset $U$ of $\PP^1
(\C_p)$.