In an answer to [this question](http://math.stackexchange.com/q/743985/3217) I was led to show the  trick proving that $\mathbb R$ is the fraction field of some strict subring $A\subsetneq \mathbb R=\operatorname{Frac}(A)$.  
A crucial point in the trick  is  the existence of a transcendence basis of $\mathbb R$ over  $\mathbb Q$.  
So this necessitates the axiom of choice and I would be grateful to a savvy user for confirming (or infirming) my feeling  that the existence of such an $A$ is indeed only possible under acceptance of the axiom of choice.