Show that:
1. The additive group is isomorphic to the multiplicative group of all complex numbers of modulus 1.
2. The additive group is isomorphic to the multiplicative group of all complex roots of unity.
Okay, so I understand that I need find a mapping between these groups, prove they are well-defined, linear, and one to one and onto, but would anyone please explain to me how the complex numbers of mod 1 and roots of unity going to behave?
Complex numbers of mod 1, wouldn't that be any combination of a(i)?
Roots of unity, well, I know how it would work in integers, but complex, I'm a bit lost on that.
Thanks for all your help!!!
let be the multiplicative group of all complex numbers of modulus 1. define by clearly is a homomorphism. now if then
so is onto. also so
let be the the multiplicative group of all complex roots of unity. define by the same as above is a homomorphism and now choose then we have2. The additive group is isomorphic to the multiplicative group of all complex roots of unity.
for some integers hence so is onto. Q.E.D.