2 results
Abelian absolute Galois groups: In Erinnerung an Wulf-Dieter Geyer (1939–2019)
- Part of
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- Journal:
- Glasgow Mathematical Journal / Volume 66 / Issue 2 / May 2024
- Published online by Cambridge University Press:
- 02 February 2024, pp. 359-367
- Print publication:
- May 2024
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- Article
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Generalizing a result of Wulf-Dieter Geyer in his thesis, we prove that if
$K$ is a finitely generated extension of transcendence degree 
$r$ of a global field and 
$A$ is a closed abelian subgroup of 
$\textrm{Gal}(K)$, then 
${\mathrm{rank}}(A)\le r+1$. Moreover, if 
$\mathrm{char}(K)=0$, then 
${\hat{\mathbb{Z}}}^{r+1}$ is isomorphic to a closed subgroup of 
$\textrm{Gal}(K)$.
THE ELEMENTARY THEORY OF LARGE FIELDS OF TOTALLY
$\mathfrak{S}$ -ADIC NUMBERS
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 16 / Issue 1 / February 2017
- Published online by Cambridge University Press:
- 23 April 2015, pp. 121-154
- Print publication:
- February 2017
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We analyze the elementary theory of certain fields
$K^{\mathfrak{S}}(\boldsymbol{\unicode[STIX]{x1D70E}})$ of totally 
$\mathfrak{S}$ -adic algebraic numbers that were introduced and studied by Geyer and Jarden and by Haran, Jarden, and Pop. In particular, we provide an axiomatization of these theories and prove their decidability, thereby giving a common generalization of classical decidability results of Jarden and Kiehne, Fried, Haran, and Völklein, and Ershov.
