Specifically, I've been reading a standard procedure to solve a general cubic polynomial by taking field extensions by radicals.
As part of the procedure, a cube root $\omega$ of unity is adjoined. At first, this step seemed fine to me because it is all about adjoining radicals. But then I started thinking about how the mathematicians in the 1500's posed the problem. Take the coefficients of a polynomial, and then use arithmetic operations and root extraction to provide a formula to solve the polynomial. Adjoining a cube root $\omega$ for a field extension seems to provide extra support, above and beyond what is "allowed" to solve a polynomial.
Justify adjoining roots of unity to solve a polynomial by radicals. That is, show that solving a polynomial by taking field extensions by radicals, including roots of unity, still leads to an explicit formula using arithmetic operations and root extraction on the coefficients.