Suppose we consider a rigid extension field $F$, i.e., $\text{Aut}(F) = 1$ over the complex numbers $\mathbb C$. What is the minimal cardinality of $F$? In particular it should hold that in this case $|F| > |\mathbb C|$.

Moreover, if we replace $\mathbb C$ with any other algebraically closed field, what can one say in this case?

Any comment, reference, or pointer is highly appreciated.

All the best, Sebastian