Ghiloni, Riccardo (2009) The classification of algebraically closed alternative division rings of finite central dimension. UNSPECIFIED. (Unpublished)
Abstract
A classical result of Noncommutative Algebra due to I. Niven, N. Jacobson and R. Baer asserts that an associative noncommutative division ring D has finite dimension over its center R and is algebraically closed (that is, every nonconstant polynomial in one indeterminate with left, or right, coefficients in D has a root in D) if and only if R is a real closed field and D is isomorphic to the ring of quaternions over R. In this paper, we extend this classification result to the nonassociative alternative case: the preceding assertion remains valid by replacing the quaternions with the octonions. As a consequence, we infer that a field k of characteristic not 2 is real closed if and only if the ring of octonions over k is an algebraically closed division ring.
| Item Type: | Departmental Technical Report |
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| Department or Research center: | Mathematics |
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| Subjects: | Q Science > QA Mathematics > QA152 Algebra |
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| Uncontrolled Keywords: | Algebraically closed rings, Alternative division rings, Real closed fields, Quaternions, Octonions |
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| Report Number: | UTM 729, June 2009 |
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| Repository staff approval on: | 23 Jun 2009 |
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