Exercise 1.6.4 (The multiplicative groups $\mathbb{C}^*$ and $\mathbb{R}^*$ are not isomorphic)

Question

Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

\begin{proof} We write $\R^* = \mathbb{R} - \{0\}$ and $\C^* = \mathbb{C} - \{0\}$.

Put $\omega = e^{2i\pi/3} = \frac{-1}{2} + i \frac{\sqrt{3}}{2}$. Then $\omega 
e 1$ and $\omega^3 = 1$, so the order of $\omega$ in the group $(\C^*, 	imes)$ is $3$.

But $\R^*$ has no element $a$ of order $3$, otherwise $a
e 0, a 
e 1$ and $a^3 = 1$, so $(a-1)(a^2 + a + 1) = 0$, with $a
e 1$ thus
$$0 = a^2 + a + 1 = \left( a+ \frac{1}{2} ight)^2 + \frac{3}{4} >0.$$
This gives the contradiction $0 >0$, which proves that the group  $\R^*$ has no element of order $3$.

By Exercise 2, this shows that the groups $(\R^* , 	imes)$ and $(\C^*, 	imes)$ are not isomorphic.

\end{proof}

Exercise 1.6.4 (The multiplicative groups $\mathbb{C}^$ and $\mathbb{R}^$ are not isomorphic)

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Exercise 1.6.4 (The multiplicative groups $\mathbb{C}^*$ and $\mathbb{R}^*$ are not isomorphic)

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Exercise 1.6.4 (The multiplicative groups $\mathbb{C}^$ and $\mathbb{R}^$ are not isomorphic)