No transcendence basis of R over Q can be analytic

Abstract

It has been proved by Sierpiński that no linear basis of $\mathbb{R}$ over $\mathbb{Q}$ can be an analytic set. Here we show that the same assertion holds by replacing ``linear basis'' with ``transcendence basis''. Furthermore, it is demonstrated that purely transcendental subfields of $\mathbb{R}$ generated by Borel bases of the same cardinality are Borel isomorphic (as fields). Following Mauldin's arguments, we also indicate, for each ordinal $\alpha$ such that $1\leq \alpha\lt\omega_1$ ($2\leq \alpha\lt\omega_1$), the existence of subfields of $\mathbb{R}$ of exactly additive (multiplicative, ambiguous) class $\alpha$ in $\mathbb{R}$.