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arXiv · 2502.15674

On rational connectedness and parametrization of finite Galois extensions

Daniel Krashen·Danny Neftin
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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 $G$.

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Daniel Krashen, Danny Neftin. 2025-02-21. On rational connectedness and parametrization of finite Galois extensions. https://arxiv.org/abs/2502.15674

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