Two Exercises Concerning the Degree of the Product of Algebraic Numbers


Let $k$ be a field, and let $\alpha$ and $\beta$ be two algebraic numbers over $k$ of degree $d$ and $\ell$, respectively. We find necessary and sufficient conditions under which $\deg(\alpha\beta)=d\ell$ and $\deg(\alpha+\beta)=d\ell$. Since these conditions are quite difficult to check, we also state a simple sufficient condition for such equalities to occur.