[cdfncffr] Algebraic number coefficients

An algebraic number is typically defined as a root of a polynomial with integer (or rational) coefficients.  Surprisingly, a polynomial with algebraic number coefficients also has roots that are algebraic numbers; there are no "second-order" algebraic numbers.  Is the proof of this constructive?

Consider a "parent" polynomial whose coefficients are given as the roots of explicitly specified "child" polynomials with integer coefficients.  Let X be a root of the parent polynomial.  Express X as the root of a polynomial with coefficients that are integers.  These integers would have to be a function of the integer coefficients of the child polynomials.

Things might get messy as a polynomial typically has many roots, so we need a way of specifying a particular one.