Division algebras with dimension 2 t , t ∈ N

In this paper we find a field such that the algebras obtained by the Cayley-Dickson process are division algebras of dimension 2t,∀t ∈ N. Subject Classification: 17D05; 17D99. From Frobenius Theorem and from the remark given by Bott and Milnor in 1958, we know that for n ∈ {1, 2, 4} we find the real division algebras over the real field R. These are: R, C, H(the real quaternion algebra), O(the real octonions algebra ). They are unitary and alternative algebras. In 1978, Okubo gave an example of a division non alternative and non unitary real algebra with dimension 8, namely the real pseudo-octonions algebra.(See[7]). Here we find a field such that the algebras obtained by the Cayley-Dickson process are division algebras of dimension 2, ∀t ∈ N. First of all we describe shortly the Cayley-Dickson process. Definition 1. Let U be an arbitrary algebra. The vector spaces morphism φ : U → U is called an involution of the algebra U if φ (φ (x)) = x and φ (xy) = φ (y)φ (x) , ∀x, y ∈ U. Let U be a arbitrary finite dimensional algebra with unity, 1 = 0, with an involution φ : U → U, φ (a) = a,where a+ a and aa belong in K · 1, for all a in U . Let α ∈ K, be a non zero fixed element . Over the vector space U ⊕ U, we define the multiplication: (a1, a2) (b1, b2) = ( a1b1 − αb2a2, a2b1 + b2a1 ) . (1)