This question is related to [Degree of sum of algebraic numbers](https://mathoverflow.net/questions/26832/degree-of-sum-of-algebraic-numbers). Forgive me if
this is a dumb question, but are there two algebraic numbers $a$ and $b$ of degree $3$ and $6$ respectively, such that the sum $a+b$ has degree $12$ ?


Intuitively it would seem that the degree of $a+b$ should divide $3 \times 6=18$, but I was unable to prove this. Hence my question. 



algebraic numbers of degree 3 and 6, whose sum has degree 12
The splitting field $K$ of $x^6-2$ has degree 12 over the rationals. Its Galois group is the dihedral group 
$D\_{12}$. This group has subgroups $A$ and $B$, where $A$ has order 2, $B$ has order 4, and $B$ does not contain 
$A$; the subgroup generated by $A$ and $B$ together is all of $D\_{12}$. Now let $E$ and $F$ be the fixed fields of $A$ and $B$ respectively; then $E$ has degree 6, $F$ has degree 3, and $F$ is not contained in $E$, so the smallest field containing both $E$ and $F$ is $K$. Now if you pick pretty much any $a$ in $F$ and pretty much any $b$ in $E$, you should have what you want. 


