$1.$ Let $f(x) =a_mx^m+a_{m-1}x^{m-1}+ \cdots +a_0$ and $g(x) = b_nx^n+b_{n-1}x^{n-1}+ \cdots +b_0$ belong to $\mathbb Q[x]$ and suppose that $f \circ g \in \mathbb Z[x].$ Prove that $a_ib_j$ is an integer for every $i,j$
Attemt: Given that $f(x), g(x) \in Q[x], ~~f \circ g \in \mathbb Z[x] $
Then $f \circ g (x)= f(g(x)) = a_mg^m(x)+a_{m-1} g^{m-1}(x) + \cdots +a_0$
Now, $g^i(x) = \sum_{k=1}^{i} \sum_{l=1}^k b_1 b_2 \cdots b_l x^k $ such that $1+2+ \cdots +l=k$
$f \circ g (x) = \sum_{i=1}^{mn}~ \sum_{k=1}^{i} \sum_{l=1}^k a_k b_1 b_2 \cdots b_l x^k $
The final coeffecients obtained after multiplication and summation belong to $\mathbb Z$. But what does that say about the individual combinations $a_ib_j?$
This method seems not feasible much.. Hence, If we think like this : $Q[x]$ forms a principal ideal domain.$\implies f(x) = \langle f_1(x) \rangle $ and $g(x)= \langle g_1(x) \rangle $
How do I proceed ahead?
$2.$ Let $g(x)$ and $h(x)$ belong to $\mathbb Z[x]$ and let $h(x)$ be monic. If $h(x)$ divides $g(x)$ in $\mathbb Q[x]$, then prove that $h(x)$ divides $g(x)$ in $\mathbb Z[x]$
Attempt: $g(x) = h(x) p(x)~|~p(x) \in \mathbb Q[x] \implies g(x) \in \langle h(x) \rangle $
$h(x)$ is monic $\implies$ if the highest coeffecient of $g(x) =a$, then if $\deg h(x) = m, \deg g(x) = n ; n>m :$
$\deg p(x) =n-m$ and leading coeffecient of $p(x)=a$.
How do I proceed ahead now?
Thank you for your help. Please note that my book has not yet introduced factorization of polynomials nor reducability, unique factorization domain.