Is the following proposition true? If yes, how would you prove this?
Proposition Let $K$ be an algebraic number field. The group of roots of unity in $K$ is finite. In other words, the torsion subgroup of $K^*$ is finite.
Motivation Let $A$ be the ring of algebraic integers in $K$. A root of unity in $K$ is a unit(i.e. an invertible element of $A$). It is important to determine the structure of the group of units in $K$ to investigate the arithmetic properties of $K$.
Remark Perhaps, the following fact can be used in the proof. Every conjugate of a root of unity in $K$ has absolute value 1,
Edit(Jan. 18, 2013) To the downvoters, why don't you reset your votes? The question is clearly important in algebraic number theory. I'm saying this not because I care my reps, but because the negative votes are sending wrong signals to the users.