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Topological Realisations of Absolute Galois Groups

  • Robert A. Kucharczyk
  • Peter ScholzeEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 245)

Abstract

Let F be a field of characteristic 0 containing all roots of unity. We construct a functorial compact Hausdorff space \(X_F\) whose profinite fundamental group agrees with the absolute Galois group of F, i.e. the category of finite covering spaces of \(X_F\) is equivalent to the category of finite extensions of F. The construction is based on the ring of rational Witt vectors of F. In the case of the cyclotomic extension of \(\mathbb {Q}\), the classical fundamental group of \(X_F\) is a (proper) dense subgroup of the absolute Galois group of F. We also discuss a variant of this construction when the field is not required to contain all roots of unity, in which case there are natural Frobenius-type automorphisms which encode the descent along the cyclotomic extension.

Keywords

Galois groups Fundamental groups Witt vectors 

2010 Mathematics Subject Classification

12F10 11R32 14F35 

Notes

Acknowledgements

Part of this work was done while the second author was a Clay Research Fellow. All of it was done while the first author was supported by the Swiss National Science Foundation. The authors wish to thank Lennart Meier for asking a very helpful question, Markus Land and Thomas Nikolaus for a discussion about Proposition 7.10, and Eric Leichtnam for pointing out some typographical errors in an earlier version.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département MathematikETH ZürichZürichSwitzerland
  2. 2.Mathematisches InstitutRheinische Friedrich-Wilhelms-Universität BonnBonnGermany

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