I read years ago in the great French book "l'aventure des nombres" (Gilles Godefroy, Odile Jacob, 1997) that the axiom of determinacy was incompatible with the axiom of choice. The latter seems to be needed to allow the existence of discontinuous field automorphisms of $C$. As some of you may know, I have been working for some time on an approach of RH based on automorphisms of rigs (rings without negative) whose maximal one I conjecture to correspond to "the" algebraic closure of $Q$.

In other words, denoting by $\mathcal{M}$ the maximal "L-rig", i.e. rig whose elements are L-functions, I expect $\operatorname{Aut}(\mathcal{M})$ to be isomorphic to $\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. Defining the symmetry group $\operatorname {Sym}(F)$ of a given element $F$ of $\mathcal{M}$ as made of elements of $\operatorname {Aut}(\mathcal{M})$ preserving $F$ so as to establishing its isomorphy with the isometry group of the multiset of non trivial zeroes of $F$, the latter having to be seen as the automorphism group of the Euclidean domain where this multiset lies, i.e. the critical strip, one has to restrict the potential candidates in $\operatorname{Aut}(\mathcal{M})$ to elements of finite order, i.e. either the identity or conjugates of the complex conjugation (as elements of finite order of $\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$), or continuous elements of this absolute Galois group, i.e. the identity and the complex conjugation. Such a restriction seems rather artificial, and is not quite satisfying. So, my question is:

Would working with the axiom of determinacy instead of the axiom of choice provide a more natural framework for RH? Has this conjecture been proven to be provable in ZF without the requirement of the axiom of choice?