Prove that $\mathbb{R}^\times\times\mathbb{R}^\times\ncong\mathbb{C}^\times$

Question

As the title says, prove that the direct product of real number with real number with respect to multiplication is not isomorphic to complex numbers with respect to multiplication. I proved in the last part that $\mathbb{R}\times\mathbb{R}\cong\mathbb{C}$, but I don't know how to progress to this part. Please help. Thanks in advance.

Answer 1

Find an element of $\mathbb{C}$ that behaves differently under multiplication than any real number. Like $\mathfrak{i}$, perhaps. What properties does $\mathfrak{i}$ have under multiplication? Is there an element of $\mathbb{R} \times \mathbb{R}$ that behaves like $\mathfrak{i}$? Nope.

Answer 2

One option is to think about how many elements satisfy the equation $x^2 = 1$. (The fancy name for this is to consder the ''two-torsion'' of the multiplicative groups.)

In $\mathbb C$ there are two solutions, $x = 1$ and $-1$. In $\mathbb R \times \mathbb R$ there are more -- do you see why?

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You can also try the same thing with 3-torsion: how many elements satsify $x^3 = 1$ in each case?