Mahler measure of the Horie unit and Weber's class numberproblem in the cyclotomic Z_3-extension of Q

摘要

Let p be a prime number and Q(μ_p∞) the cyclotomic field of all p-power roots of unity. Let B_(p,n) be the unique real subfield of Q(μ_p∞) which is cyclic of degree p~n over Q. Then we call B_(p,∞)=∪ _(n≥1)B_(p,n) the cyclotomic Z_p-extension of Q, and B_(p,n) the nth layer of this extension. We denote the class number of B_(p,n) by h_(p,n). We consider the following problem: WEBER'S CLASS NUMBER PROBLEM. Is hp,n equal to one for every positive integer n?