Simple birational extensions of the polynomial ring \(\mathbb{C}^{[3]}\) On the embedding of \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\) into \(\widehat{GT}\). (Appendix by Michel Emsalem and Pierre Lochak: The action of the absolute Galois group on the moduli space of spheres with four marked points) 