If $p_1,\ldots,p_n$ are polynomials over a field $F$, not all of which are $0$, there is a unique monic polynomial $d$ in $F[x]$ such that
(a) $d$ is in the ideal generated by $p_1, \ldots, p_n$;
(b) $d$ divides each of the polynomials $p_i$.
Any polynomial satisfying (a) and (b) necessarily satisfies
(c) $d$ is divisible by every polynomial which divides each of the polynomials $p_1,\ldots,p_n$.
I want to prove this theorem, but I don't understand the proof in my linear algebra text. Also, I want to prove "$d$ is the monic generator of the ideal".
Actually, the text starts the proof identifying $d$ as the monic generator without any proof. Even if the proof is same here but the verification that $d$ is the monic generator is in it, I can understand the proof.