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Two exercises concerning the degree of the product of algebraic numbers. (English) Zbl 1220.11131
Summary: 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.

##### MSC:
 11R04 Algebraic numbers; rings of algebraic integers 11R32 Galois theory 12E99 General field theory
##### Keywords:
algebraic numbers; degree; root of unity
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