What does it mean that "the Complex conjugation is a Field-automorphism of ${\mathbb {C} }\,$'?

Question

I was doing a test and in one point I had to show that the following is true:

(1) $ {\overline {a_1}-\overline {a_2}} = \overline {a_1-a_2} $

(2) ${\overline{{a_1}{a_2}-{b_1}\overline {b_2}}} = \overline {a_1}\overline {a_2}-\overline{b_1}b_2$

It was a bit long, but I proved it using the form $z = x+yi.$

Although, I also check the official solution, and there it was written: 'those are true, because the Complex conjugation is a Field-automorphism of ${\mathbb {C} }\,$'

What does this sentence mean. How one can prove that the Complex conjugation is a Field-automorphism of ${\mathbb {C} }\,$?

Thank you for your help in advance.

Answer 1

It means that it is a rearrangement of the complex numbers in such a way that $+$ and $\times$ still work right.

$\mathbb{C}$ = the complex numbers

field = a system of numbers, such as $\mathbb{C}$ (or the real numbers $\mathbb{R}$, or the rational numbers $\mathbb{Q}$), and two operations $+$ and $\times$ that obey the rules you are used to, such commutativity and associativity. (Here is the Wikipedia entry.)

field automorphism = a switching around of the numbers in a field in such a way that sums get sent to sums and products get sent to products. In other words, a (bijective) map $f : \mathbb{C}\rightarrow\mathbb{C}$ such that $f(a+b) = f(a) + f(b)$ and $f(ab) = f(a)f(b)$ for all $a$ and $b$ in the field

So complex conjugation is a field automorphism because $\overline{a+b} = \overline{a} + \overline{b}$ and $\overline{ab} = \overline{a}\overline{b}$.

The idea of a field automorphism is that $+$ and $\times$ define the structure of the field, so an automorphism is a rearrangement of the elements of the field that "preserves its structure."