The Generating Degree of ℂ_<i>p</i>

Abstract

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 [ M ] is dense in A . For a prime number p , denotes the topological completion of an algebraic closure of the field of p -adic numbers. We prove that gdeg( ) = 1, i.e., there exists t in such that [t] is dense in . We also compute where A(U) is the ring of rigid analytic functions defined on a ball U in . If U is a closed ball then = 2 while if U is an open ball then is infinite. We show more generally that is finite for any affinoid U in ℙ 1 ( ) and is infinite for any wide open subset U of ℙ 1 ( ).

Locations

• Canadian Mathematical Bulletin - View - PDF