Functions like Factor usually assume that all coefficients in the polynomials they produce must involve only rational numbers. But by setting the option Extension you can extend the domain of coefficients that will be allowed.
factor poly allowing algebraic numbers in poly to appear in coefficients
Factoring polynomials with algebraic number coefficients.
Here is a polynomial with a coefficient involving :
By default, Factor will not factor this polynomial:
But now the field of coefficients is extended by including , and the polynomial is factored:
Other polynomial functions work much like Factor. By default, they treat algebraic number coefficients just like independent symbolic variables. But with the option Extension->Automatic they perform operations on these coefficients.
By default, Cancel does not reduce these polynomials: