The number of roots of unity in an imaginary quadratic number field The book I am studying uses the notation $w_k$ to denote the number of roots of unity contained in $K$, an imaginary quadratic number field.
Supposedly, this is the same as the size of the unit group in $O$, the ring of integers in $K$.  I am well familiar with the standard arguments for finding the units in $O$.  
Im not really sure exactly what the phrase "the number of roots of unity in $K$" means precisely.  Does anyone know?
Also, I was thinking that it is probibly the case that the only elements of $K$ with finite multiplicative order are the elements of the unit group.  Is this correct?  If so, then is this the correct line of reasoning to show that $w_k=|U(O)|$?
 A: A root of unity is a complex number $a$ such that $a^n = 1$ for some positive integer $n$. The "number of roots of unity" is the cardinality of the set of roots of unity. 
For example, if $K=\mathbb{Q}(i)$, then the roots of unity in $K$ are $1$, $-1$, $i$, and $-i$, so the number of roots of unity contained in $K$ is $4$.
Yes: a complex number is a root of unity if and only if it has finite multiplicative order in $\mathbb{C}^*$. In fact, the group of roots of unity is the torsion subgroup of $\mathbb{C}^*$. 
To prove that in the imaginary quadratic case the unit group of $\mathcal{O}_K$ coincides with the group of roots of unity in $K$ you could use Dirichlet's Unit Theorem (as Timothy Wagner suggests), but you don't have to. You can simply show that every unit in $\mathcal{O}_K$ must in fact be a root of unity. This is not hard, since you (probably) know exactly what $\mathcal{O}_K$ for $K=\mathbb{Q}(\sqrt{d})$ ($d$ square free, $d\neq 1$), and you can use the norm map. For example, to show it in the case of $K=\mathbb{Q}(i)$, note that an element in $\mathcal{O}_K = \mathbb{Z}[i]$ is a unit if and only if $N(a+bi) = a^2+b^2 = 1$ (since $N((a+bi)(c+di)) = N(a+bi)N(c+di)$). But since $a$ and $b$ are integers, that requires $a=\pm 1$ and $b=0$ or $a=0$ and $b=\pm 1$, giving you roots of unity. 
(In fact, imaginary quadratic fields are the only number fields in which the group of units in the number field is finite and torsion; this follows from the aforementioned Dirichlet Unit Theorem)
A: You need Dirichlet's unit theorem.
http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/unittheorem.pdf
