In Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which:
- there are vector spaces without a basis;
- the field of complex numbers $\mathbb{C}$ only has two automorphisms (identity and complex conjugation).
Before I even ask my main question, I want to ask two "pre-questions" (as I am not a logician):
- is my formulation "in Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which ..." formally correct? (And if not, what is a formal correct statement ?);
- for the second statement above, what is a precise reference in which I can find this statement?
Now for my main question: is it also true that it is consistent to say that there are models of ZF set theory without AC, in which every vector space over $\mathbb{C}$ has a base (or even stronger: in which dimension is well defined), and in which $\mathbb{C}$ also has precisely two field automorphisms?
