## Abstract

Let D be a division algebra with center F. Consider the group CK _{1}(D) = D*/F*D′ where D* is the group of invertible elements of D and D′ is its commutator subgroup. In this note we shall show that, assuming a division algebra D is a product of cyclic algebras, the group CK^{1}(D) is trivial if and only if D is an ordinary quaternion algebra over a real Pythagorean field F. We also characterize the cyclic central simple algebras with trivial CK_{1} and show that CK _{1} is not trivial for division algebras of index 4. Using valuation theory, the group CK_{1}(D) is computed for some valued division algebras.

Original language | English |
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Pages (from-to) | 1427-1435 |

Number of pages | 9 |

Journal | Communications in Algebra |

Volume | 33 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2005 |

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