Abstract algebra help?

Let f: R -%26gt; C* be the map given by f(r) = cos(2pir) + i*sin(2pir).





a. Prove that f is homomorphism.


b. What is Kernel(f)?


c. What is Im(f)?


d. To what group is R/kernel(f) isomorphic?

Abstract algebra help?
f(r) = e^(2 pi r i)





(a) Law of exponents gives f(x+y) = f(x) f(y)





(b) For what r is f(r)= 1? You must have


cos (2 pi r) = 1 and sin(2 pi r) = 0 and that happens for integers r. The kernel is Z, the additive group of integers.





(c) Image is all of the unit circle of complex numbers, the numbers of absolute value 1.





(d) R/Kernel is isomorphic to the image figured out in (c)). It's a circle group.
Reply:It has been a while since I have taken this class. Go to www.cramster.com





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