Questions:
Prove that the fields R and C are not isomorphic.
Proof:
Proving by Contradiction
Temporary assume they are isomorphic, then there exists a ring isomorphism g: C→R
g(a.b) = g(a).g(b) for all a,b ∈ C
= g[(i).(-i)] for all i ∈ C
= g(i).g(-i)
= -g(i).g(i)
= -g(i)2
Then g(1) = 1
g(1) = -g(i)2
1 = -g(i)2
-1 = g(i)2
The contradiction is when -1 = g(i)2 because g(i)2 ∈ R but -1 ∉ R. Therefore, R and C are not isomorphic.
Need to justify three things:
1) g(i)???
2) g(i).g(-i)
= -g(i).g(i) why?
3) g(1) = 1 why?