Let $A,B$ be two nonzero polynomials, I want to show that there exists a monic polynomial $P$ such that $D(A,B)=D(P)$ (with $D(P)=\{Q\in\mathbb{R}: Q|P\}$).
For this I considered the set $E=\{\deg(P):P\in D(A,B)\}$ and is a non-empty set of $\mathbb{N} $ bounded above therefore admits a maximum, we set $n_0=\max(E)$.
Let $P\in D(A,B)$ be unitary such that $n_0=\deg(P)$ then $D(P)\subset D(A,B)$. But I have a problem to show the other inclusion. An idea please.
