Remark

The crux of the matter is the following fact: Given two polynomials f, g Q[X], there are polynomials h, k Q[X], such that

• the sum of each root of f and each root of g is a root of h,
• the product of each root of f and each root of g is a root of k.

The proof of these statements is beyond the scope of these notes. But constructions of such polynomials were given in several examples.