The Galois group of a Special Group

Kaique Matias de Andrade Roberto; Hugo Luiz Mariano

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arXiv · 2404.03785

The Galois group of a Special Group

Kaique Matias de Andrade Roberto·Hugo Luiz Mariano

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Abstract

In this ongoing work, we extend to a class of well-behaved pre-special hyperfields the work of J. Min\'a\v c and Spira (\cite{minac1996witt}) that describes a (pro-2)-group of a field extension that encodes the quadratic form theory of a given field $F$: in \cite{adem1999cohomology} it is shown that its associated cohomology ring contains a copy of the cohomology ring of the field $F$. Our construction, a contravariant functor into the category of "pointed" pro-2-groups, is essentially given by generators and relations of profinite-2-groups. We prove that such profinite groups $\mbox{Gal}(F)$ encode the space of orders of the special group canonically associated to the hyperfield $F$ and provide a criterion to detect when $F$ is formally real or not.

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Kaique Matias de Andrade Roberto, Hugo Luiz Mariano. 2024-04-04. The Galois group of a Special Group. https://arxiv.org/abs/2404.03785

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