Is it known what is the centralizer of the complex conjugation in the absolute Galois group (i.e. the Galois group of the field of complex algebraic numbers over the rationals)? and, what would be a good reference for this question?



Centraliser of the complex conjugation in the absolute Galois group
If some element centralizes the complex conjugation, then it must preserve the real numbers as a set. Now, since any automorphism of the real numbers preserves the set of squares, it must preserve the order; and hence be continuous. Since $\mathbb Q$ is fixed, this implies that the real numbers are fixed pointwise. It follows that any element which centralized the complex conjugation must be the identity or the complex conjugation itself.


