||We will talk about a lower bound for Weil's height. The original question is "can we find an algebraic number which has height very close to 0 in a fixed finite extension L of Q"? In this talk, we will show that if L is an n-th cyclotomic extension of Q, then the answer of the question is negative. Furthermore, by Kronecker-Weber's theorem, we may assume that L is an abelian extension of Q. Via our main result, it will be a tool to estimate the lower bound of norm and to list all cyclotomic fields with class number 1.