The Classification of Algebraically Closed Alternative Division Rings of Finite Central Dimension

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.