Functiones et Approximatio Commentarii Mathematici

The absolute Galois group of subfields of the field of totally $\boldsymbol{S}$-adic numbers

Dan Haran, Moshe Jarden, and Florian Pop

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For a finite set $S$ of local primes of a countable Hilbertian field $K$ and for $\sigma_1,\ldots,\sigma_e\in\Gal(K)$ we denote the field of totally $S$-adic numbers by $\K_{tot,S}$, the fixed field of $\sigma_1,\ldots,\sigma_e$ in $\K_{tot,S}$ by $\K_{tot,S}({\mathbf \sigma})$, and the maximal Galois extension of $K$ in $\KtotS({\mathbf \sigma})$ by $\KtotS[{\mathbf \sigma}]$. We prove that for almost all ${\mathbf \sigma}\in\Gal(K)^e$ the absolute Galois group of $\K_{tot,S}[{\mathbf \sigma}]$ is isomorphic to the free product of $\hat{F}_\omega$ and a free product of local factors over $S$.

Article information

Funct. Approx. Comment. Math., Volume 46, Number 2 (2012), 205-223.

First available in Project Euclid: 25 June 2012

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Zentralblatt MATH identifier

Primary: 12E30: Field arithmetic

Hilbertian field local primes totally $S$-adic numbers Haar measure absolute Galois group free product


Haran, Dan; Jarden, Moshe; Pop, Florian. The absolute Galois group of subfields of the field of totally $\boldsymbol{S}$-adic numbers. Funct. Approx. Comment. Math. 46 (2012), no. 2, 205--223. doi:10.7169/facm/2012.46.2.6.

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