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I am trying to show that ##\mathbb{R} - \{ 0\}## is not isomorphic to ##\mathbb{C} - \{0 \}##. If we simply look at ##x^3 = 1##, it's clear that ##\mathbb{R} - \{ 0\}## has one solution while ##\mathbb{C} - \{0 \}## has three.
My question, how can I use ##x^2 = -1## to show that they are not isomorphic? Using ##x^3 = 1## is more clear because any isomorphism would preserve powers and preserve the identity. But using ##x^2 = -1## is less clear to me.
My question, how can I use ##x^2 = -1## to show that they are not isomorphic? Using ##x^3 = 1## is more clear because any isomorphism would preserve powers and preserve the identity. But using ##x^2 = -1## is less clear to me.