## Real Analysis Exchange

### No transcendence basis of R over Q can be analytic

Enrico Zoli

#### 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 $\mathbb{R}$.

#### Article information

Source
Real Anal. Exchange, Volume 30, Number 1 (2004), 311-318.

Dates
First available in Project Euclid: 27 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.rae/1122482136

Mathematical Reviews number (MathSciNet)
MR2127535

Zentralblatt MATH identifier
1061.28001

#### Citation

Zoli, Enrico. No transcendence basis of R over Q can be analytic. Real Anal. Exchange 30 (2004), no. 1, 311--318. https://projecteuclid.org/euclid.rae/1122482136