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# homomorphism watch

1. just a check ive got this correct.

asked to show the map from complex numbers under multiplication given by

Z maps to Z*/Z is a homomorphism and give its image and kernel

first bit no problem

for kernel I have said it is set of all z such that z*/z=1 so z*=z so ker is the real numbers

for image I have said its set of w such that w=z*/z for some z and this is just all w with modulus 1 so on unit circle

then have used iso thereom to get

C*/R isomorphic to unit circle in z plane

is this correct??
2. (Original post by mathz)
just a check ive got this correct.

asked to show the map from complex numbers under multiplication given by

Z maps to Z*/Z is a homomorphism and give its image and kernel

first bit no problem

for kernel I have said it is set of all z such that z*/z=1 so z*=z so ker is the real numbers

for image I have said its set of w such that w=z*/z for some z and this is just all w with modulus 1 so on unit circle

then have used iso thereom to get

C*/R isomorphic to unit circle in z plane

is this correct??
Could you post a picture of the question, please?
3. (Original post by mathz)
just a check ive got this correct.

asked to show the map from complex numbers under multiplication given by

Z maps to Z*/Z is a homomorphism and give its image and kernel

first bit no problem

for kernel I have said it is set of all z such that z*/z=1 so z*=z so ker is the real numbers

for image I have said its set of w such that w=z*/z for some z and this is just all w with modulus 1 so on unit circle

then have used iso thereom to get

C*/R isomorphic to unit circle in z plane

is this correct??
Post a question because it is hard to understand what you are asking but here is the theorem it looks like you want to apply.
Let be rings and be a homomorphism, then

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