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Prove that the fields R and C are not isomorphic.

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Check if the proof is correct. I need help to justify some of my answers by using Theorems, Definitions and etc. You can change my wording but try to stick to my idea. It's really important that you explain your work. Thanks!

Note: R=Real Number and C=Complex Number

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It is proven that the fields R and C are not isomorphic.

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Your proof is correct. I verify something in your proof with respect to your request.
g: C -> R is a ring homomorphism, then we have
g(x+y) = g(x) + g(y)
g(xy) = g(x)g(y)
g(0) = 0

(1) First, I claim that g(1) = 1.
Since g is an isomorphism, ...

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