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I'm doing Exercise I.11.8 from textbook Analysis I by Amann/Escher.

Show that the identity function and $z \mapsto \overline{z}$ are the only field automorphisms of $\mathbb{C}$ which leave the elements of $\mathbb{R}$ fixed.

Could you please verify if my attempt contains logical gaps/errors?


My attempt:

Let $\phi:\mathbb{C} \to \mathbb{C}$ be an automorphism of $\mathbb{C}$ which leaves the elements of $\mathbb{R}$ fixed.

Since group homomorphism maps identity elements to identity elements, $\phi(1) = 1$ and $\phi(0) = 0$. Then $(\phi(i))^2 = \phi(i^2) = \phi (-1) = - \phi(1) = -1$, so $\phi(i) = \pm i$.

For $z = a + ib \in \mathbb{C}$, we have $$\begin{aligned}\phi(z) &= \phi (a + ib) \\ &= \phi(a) + \phi(i) \phi(b) \\ &= a + \phi(i)b \\ &= a \pm ib \\ & = z \text{ or } \overline{z}\end{aligned}$$

This completes the proof.

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  • $\begingroup$ If you prove $\phi(1)=1$, then shouldn't you also prove $\phi(-z) = -\phi(z)$ before you use it? Or maybe just prove $\phi(-1)=-1$ without using $\phi(-z) = -\phi(z)$. Or even just prove $\phi(i) = \pm i$ directly without using $\phi(-1)=-1$. $\endgroup$ – GEdgar Jun 29 at 15:33
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It's fine, though maybe the write-up is a little terse. Your working could potentially be interpreted that, for all $z \in \Bbb{C}$, $\phi(z) = z$ or $\overline{z}$, depending on the value of $z$. That is, it might be true that, for the same $\phi$, there could exist $z$ and $w$ such that $\phi(z) = z$ and $\phi(w) = \overline{w}$.

Of course, your logic doesn't actually allow for this possibility. You're actually showing that $\forall z \in \Bbb{C}, \phi(z) = z$ OR $\forall z \in \Bbb{C}, \phi(z) = \overline{z}$, as the exercise requires. But, I think your write-up should reflect this better.

To explain this better, split into two cases: $\phi(i) = i$ and $\phi(i) = -i$. In the first case, show $\phi(z) = z$ for all $z$. In the second case, show $\phi(z) = \overline{z}$ for all $z$.

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It's not quite right. There is a typo at the end: when you wrote $=\pm z$, what was $=z\vee=\overline z$. But the main problem is this: it seems that what you proved is that, for each $z\in\mathbb C$, $\phi(z)=z\vee\phi(z)=\overline z$. But you are supposed to prove a much stronger statement:$$(\forall z\in\mathbb C):\phi(z)=z\vee(\forall z\in\mathbb C):\phi(z)=\overline z.$$That's not a serious problem, though. In fact:

  • if $\phi(i)=i$, then $(\forall z\in\mathbb C):\phi(z)=z$;
  • if $\phi(i)=-i$, then $(\forall z\in\mathbb C):\phi(z)=\overline z$.

And what you did actually proves this.

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