In this paper we prove the fundamental theorem of algebra for polynomials with coefficients in the skew field of Hamilton numbers (quaternions) and in the division algebra of Cayley numbers (octonions). The proof, inspired by recent definitions and results on regular functions of a quaternionic and of a octonionic variable, follows the guidelines of the classical topological argument due to Gauss.

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In this paper we survey a series of recent developments in the theory of functions of a hypercomplex variable. The central idea underlying these developments consists in requiring a function to be holomorphic on suitable slices of the space on which the function itself is defined. Specifically, we apply this approach to functions defined on the space H of quaternions, on the space O of octonions, and finally on the Clifford algebra of type (0, 3), denoted Cl(0, 3). The properties of these functions resemble those of holomorphic functions, and yet the different nature of the three algebras on which we work introduces new and exciting phenomena.

Let f, g be two commuting holomorphic self-maps of the unit disc. If f and g agree at the common Wolff point up to a certain order of derivatives (no more than 3 if the Wolff point is on the unit circle), then f ≡ g.1991 Mathematics Subject Classification. Primary 30A20; Secondary 30C80, 30A42.

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