Geodesic
Two Exercises Concerning the Degree of the Product of Algebraic Numbers
Publications de l'Institut Mathématique, _N_S_77 (2005) no. 91, p. 67

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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.
Classification : 11R04 11R32 12E99
Keywords: field, algebraic number, degree, root of unity 
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Arturas Dubickas. Two Exercises Concerning the Degree of the Product of Algebraic Numbers. Publications de l'Institut Mathématique, _N_S_77 (2005) no. 91, p. 67 . http://geodesic.mathdoc.fr/item/PIM_2005_N_S_77_91_a6/