Let f: R^+ maps to C^x be the map F(x)=e^ix. prove f is homomorphism & determine its kernal and image?

R is the set of all real numbers. the + means its closed under addition or its an addivitive group.


C is the set of complex numbers. x implies multiplication group. or closed under multiplication.

Let f: R^+ maps to C^x be the map F(x)=e^ix. prove f is homomorphism %26amp; determine its kernal and image?
F(x+y) = e^[i(x+y)] = e^(ix+iy) = (e^ix)(e^iy) = F(x)F(y), and thus


F is a homom.


Since |e^ix| = |cosx + isin x| = cos^2(x) + sin^2(x) = 1, the image of F is S^1 = the multiplicative sbgp. of C* of all complex numbers with module equal to 1.





Now, F(x) = 1 %26lt;==%26gt; e^ix = 1 %26lt;==%26gt; x = 2*Pi*k, with k an integer and Pi = 3.14159.... so:


Ker(F) = {2*Pi*k ; k an integer }





Regards


Tonio