Finite dimensional division algebras over a field

Date
1987
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Abstract

The problem of characterizing all division algebras over the field of real numbers has occupied algebraists since the discovery of the quaternions by Hamilton in 1843, and the discovery of the Caley numbers by Caley in 1845. In 1878, Frobenius proved that the only associative division algebras over the field of real numbers were the field of real numbers, the field of complex numbers, or the division algebra of real quaternions. Next, in 1940, Heinz Hopf showed that the dimension of any division algebra over the field of real numbers had to be a power of 2. Finally, Raul Bott and John Milnor, and independently M. Kervaire, showed that the only possible dimensions of any division algebra over the field of real numbers were 1, 2, 4, and 8. In this text, we will show that we can always find an associative division algebra of any dimension n e IN over a finite field. We will also discuss the finite dimensional associative division algebras over the field of real numbers by presenting the result of Frobenius.

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Finite fields (Algebra)
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