In the paper *Algebraische Konsequenzen des Determiniertheits-Axioms* (U. Felgner – K. Schulz, Arch. Math. (Basel) 42 (1984), pp. 557–563), the authors show that in models of Zermelo-Fraenkel set theory without the Axiom of Choice, but *with* the Axiom of Determinacy, one can show that the field of complex numbers $\mathbb{C}$ only has one nontrivial automorphism (being complex conjugation).

Can the same be said about *any* algebraically closed field in characteristic $0$ (and if so, why, or where can I found a clear reference)?

**EDIT**: in view of the answer and comments below, I want to add two more nuanced questions.

Suppose we work in Zermelo-Fraenkel without Choice. Is it consistent to say that for every field $k$ in characteristic $0$, there exists and algebraic closure $\overline{k}$ of $k$ for which the automorphism group has

*size at most $2$*?Same setting. Is it consistent to say that every uncountable algebraically closed field has at most $2$ automorphisms?