  We compare two known methods of extending a complex, unital, commutative
normed algebra so as to include solutions to sets of monic polynomials over the
original algebra. (One of these is a generalisation of a construction from the
thesis of B. J. Cole and only applies to uniform algebras.) An example is given
to show that they can lead to essentially different algebras.
  Integrally closed extensions which generalise Cole's `universal root
algebras' are also constructed .

Functional Analysis