Commutative nearly absolute valued algebras with square roots

Let $A$ be a commutative nearly absolute valued algebra. In this note we prove that all elements of $A$ have square roots if and only if $A$ is a two-dimensional real division algebra. This can be viewed as a generalization of a well-known theorem of Hopf.

Keywords

division algebra, singular homology group, commutative algebra, square root, fundamental group

2010 Mathematics Subject Classification

17A35, 46H70, 55N10

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During the realization of this work the first author was partially supported by the Ministerio de Economía y Competitividad (MTM2013-45588-C3-P and MTM2017-83506-C2-2-P).

During the realization of this work the second author was partially supported by the Ministerio de Economía y Competitividad (MTM2017-83506-C2-2-P).

The authors wish to acknowledge to professors Miguel Cabrera García and Ángel Rodríguez Palacios for the carefully reading of the manuscript and their valuable comments. They also are very grateful to the referee for his helpful suggestions. All have contributed to the improvement of this paper.

Received 27 September 2017

Received revised 12 February 2018

Published 6 June 2018