Sum, difference and product of algebraic elements is an algebraic element.

Question

I found the proof of the result "the Sum, difference and product of algebraic elements of a ring $S$ over a subring $R$, is an algebraic element over $R$", but I failed to find a polynomial in the ring $R[x]$ to show that result. Can anybody help me to find the polynomials?

Answer 1

One reason for the proof by Dedekind using finitely-generated $R$-modules is that we cannot explicitly give the polynomial for $x+y$ in terms of the polynomials for $x$ and $y$, where $x,y$ are algebraic integers. Even in easy examples it is difficult to find the polynomial. For example, let $\alpha=\sqrt{2}+\sqrt[3]{3}$. Both summands are integral, with polynomials $X^2-2$ and $X^3-3$ in $\Bbb{Z}[X]$. Do you see what the polynomial for $\alpha$ is?