This question is related to 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.