Let $D$ be an integral domain ($D$ is a noetherian UFD, if necessary) and let $a,b$ integral over $D$. Let $f$ be the minimal polynomial of $a$ over $D$ and assume it is of degree $n>1$, and let $g$ be the minimal polynomial of $b$ over $D$ and assume it is of degree $m>1$. Denote the minimal polynomial of $a+b$ over $D$ by $h$ and its degree by $d$; it is known that $1 \leq d \leq nm$.
I wish to know when $d > \min \{n,m\}$. Obviously, one has to demand that $a+b$ is not in $D$.
(Maybe adding the demand $D[a+b]= D[a+b,b]=D[a,b]$ guarantees $d \geq \min \{n,m\}$, but I am interested in $>$. Actually, I prefer not to assume $D[a+b]=D[a,b]$. Also, I do not want to demand that $n$ and $m$ are relatively prime).
Probably taking $D$ a field simplifies things, but I am interested in a noetherian UFD which is not a field.
Is my question too general (so nothing interesting can be said)?
Please notice the following somewhat relevant questions: How to prove that the sum and product of two algebraic numbers is algebraic? Sums and products of algebraic numbers
