  For a compact space X we consider extending endomorphisms of the algebra C(X)
to be endomorphisms of Arens-Hoffman and Cole extensions of C(X). Given a
non-linear, monic polynomial p in C(X)[t], with C(X)[t]/pC(X)[t] semi-simple,
we show that if an endomorphism of C(X) extends to the Arens-Hoffman extension
with respect to p then it also extends to the simple Cole extension with
respect to p. We show that the converse to this is false. For locally
connected, metric X we characterize the algebraically closed C(X), in terms of
the extendability of endomorphisms to Arens-Hoffman and to simple Cole
extensions.
