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We prove a structure theorem for the alternative finite dimensional algebras over a field K, which can be the racional numbers or an imaginary racional quadratic extension, with the hyperbolic property. One class of such algebras is the alternative totally definite octonion algebra over K. We classify the RA-loops L for which the unit loop of its integral(More)
In [4] it was shown for a suitable power n of a pair of units u, v of the quaternions algebras over the ring of integers of imaginary rational extensions A = H(o Q √ −d) that the group generated by u n , v n is a free group in the unit group of A when d ≡ 7 (mod 8) is a positive square free integer. We extend this result for any imaginary rational extension(More)
For a given division algebra of the quaternions, we construct two types of units of its Z-orders: Pell units and Gauss units. Also, if K = Q √ −d, d ∈ Z \ {0, 1} is square free and R = IK, we classify R and G such that U1(RG) is hyperbolic. In particular, we prove that U1(RK8) is hyperbolic iff d > 0 and d ≡ 7 (mod 8). In this case, the hyperbolic boundary(More)
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