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On triviality of the reduced Whitehead group over Henselian fields

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Let F be a Henselian field of q-cohomological dimension 3, where q is a prime. Let \(\Gamma _F\) be the totally ordered Abelian value group of F and let D be a central division algebra over F of index a power of q such that the characteristic of the residue field \(\overline{F}\) is coprime to q. We show that when \(1\le \dim _{{\mathbb {F}}_q}(\Gamma _F/q\Gamma _F)\le 3\), the reduced Whitehead group of D is trivial.

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Acknowledgements

The author wishes to sincerely thank J.-P. Tignol for carefully reading the manuscript and suggesting improvements. The author is grateful to A.R. Wadsworth for pointing out an error in the case \(r_q=2\) of the Theorem 1.1, in an earlier version of this manuscript. A. R. Wadsworth and J.-P. Tignol’s critical comments immensely helped in the improvement of the manuscript. Thanks are also due to B. Sury, A. Kulshrestha, and S. Shekhar for helpful discussions. We thank the reviewer for helpful suggestions.

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  1. Indian Statistical Institute, Bengaluru Center, Mysore Road, Bengaluru, India

    Abhay Soman

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  1. Abhay Soman
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Correspondence to Abhay Soman.

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The author gratefully acknowledges the support of SERB Grant (PDF/2016/000851) when part of this work was carried out.

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Soman, A. On triviality of the reduced Whitehead group over Henselian fields. Arch. Math. 113, 237–245 (2019). https://doi.org/10.1007/s00013-019-01344-3

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