Proving Q/Z isomorphic to U∗: Roots of Unity in C

[DanielThrice]

Show that Q/Z is isomorphic to the multiplicative group U∗ consisting of all roots of unity
in C. (That is, U∗ = {z ∈ C|zⁿ= 1 for some n ∈ Z+}.)

I don't really understand how to prove this isomorphism

 

[snipez90]

Use the first isomorphism/homomorphism theorem, which states that if you have a homomorphism f from G to G', then there is an isomorphism from the quotient group G/H to the image f(G), where H = Ker f.

So the idea is to exhibit a homomorphism between Q and U* whose kernel is precisely the integers. To do this, first figure out what the identity in U* is (because we need to show that our eventual homomorphism takes the integers to this identity in U*). It's really helpful in this problem if you already know precisely what the n-th roots of unity are (i.e. you know the explicit formula).