  We study the roots of polynomials over Cayley--Dickson algebras over an
arbitrary field and of arbitrary dimension. For this purpose we generalize the
concept of spherical roots from quaternion and octonion polynomials to this
setting, and demonstrate their basic properties. We show that the spherical
roots (but not all roots) of a polynomial $f(x)$ are also roots of its
companion polynomial $C_f(x)$ (defined to be the norm of $f(x)$). For
locally-complex Cayley--Dickson algebras, we show that the spherical roots of
$f'(x)$ (defined formally) belong to the convex hull of the roots of $C_f(x)$,
and we also prove that all roots of $f'(x)$ are contained in the snail of
$f(x)$, as defined by Ghiloni and Perotti for quaternions. The latter two
results generalize the classical Gauss--Lucas theorem to the locally-complex
Cayley--Dickson algebras, and we also generalize Jensen's classical theorem on
real polynomials to this setting.
