status of the complex conjugation in the complex arithmetic

Question

I am learning abstract algebra now, and get quite confused with many naive things.

A particular question is, what status does the complex conjugation have in the complex field? It is used very often in all problems. But is it a canonical or unique in some sense?

Geometrically, it is quite arbitrary---Just a reflection about the real axis. But we can also reflect about other axes, with arbitrary angles with the real axis.

score 1 · Answer 1 · answered Aug 18 '21 at 07:58

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One point is that for a polynomial $f(x)$ with real coefficients, if $z\in\Bbb C$ is a zero, then the complex conjugate $\bar z$ is also a zero, since complex conjugation is a field automorphism leaving the real numbers fixed.

answered Aug 18 '21 at 07:58

Wuestenfux

20,964

Indeed, complex conjugation is the only field automorphism of $\Bbb{C}$, other than the identity map. – Theo Bendit Aug 18 '21 at 08:59
@TheoBendit That's actually very false if one assumes the axiom of choice. There are infinitely many so-called wild automorphisms of $\mathbb{C}$. – Viktor Vaughn Aug 18 '21 at 16:10
@ViktorVaughn Oh, I thought I saw a proof of that once. I must be mistaken. Perhaps it was the only automorphism that left $\Bbb{R}$ fixed? – Theo Bendit Aug 18 '21 at 20:06
@TheoBendit Yes, since $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$ has order $2$. The identity and complex conjugation are also the only continuous automorphisms. – Viktor Vaughn Aug 19 '21 at 03:28

status of the complex conjugation in the complex arithmetic

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