On rational connectedness and parametrization of finite Galois extensions

TL;DR

This paper investigates the classification of $G$-Galois extensions of $Q$ through $R$-equivalence, determining classes for key groups and providing parametrizations without relying on a generic extension.

Contribution

It introduces the concept of $R$-equivalence for $G$-Galois extensions and characterizes these classes for fundamental groups, enabling parametrizations without a generic extension.

Findings

Classifies $R$-equivalence classes for basic groups G.

Provides explicit parametrizations of $G$-Galois extensions.

Shows how to parametrize extensions without a generic extension.

Abstract

Given two $G$-Galois extensions of $\mathbb Q$, is there an extension of $\mathbb Q(t)$ that specializes to both? The equivalence relation on $G$-Galois extension of $\mathbb Q$, induced by the above question, is called $R$-equivalence. The number of $R$-equivlance classes indicates how many rational spaces are required in order to parametrize all $G$-Galois extensions of $\mathbb Q$. We determine the $R$-equivalence classes for basic families of groups $G$, and consequently obtain parametrizations of the $G$-Galois extensions of $\mathbb Q$ in the absence of a generic extension for $G$.