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The roots of polynomials over Cayley–Dickson algebras over an arbitrary field and of arbitrary dimension are studied. It is shown that the spherical roots of a polynomial f(x) are also roots of its companion polynomial Cf(x). We generalize the classical theorems for complex and real polynomials by Gauss–Lucas and Jensen to locally-complex Cayley–Dickson algebras: it is proved that the spherical roots of f′(x) belong to the convex hull of the roots of Cf(x), and we also show that all roots of f′(x) are contained in the snail of f(x), as defined by Ghiloni and Perotti.

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... For example, when A = S with its standard basis e 0 , e 1 , . . . , e 15 (for the definition of the standard basis in terms of the generators of S and its multiplication table, see for example [5,Section 2.2] and [2, Section 2]), the elements α = e 1 + e 10 and β = e 7 + e 12 are of norm 2, so having a multiplicative norm form would mean Norm(αβ) = 4, but in fact αβ = 0. ...
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