Euclidean division of two polynomials of $\mathbb{Z}[X]$ I'm stuck on an exercice :
Let $A,B$ $\in \mathbb{Z}[X]$, where $B$ is a monic polynomial.
Consider $A = BQ + R$ the Euclidean division of $A$ by $B$ in $\mathbb{C}[X]$.
1) Show that $Q$ and $R$ are in $\mathbb{Q}[X]$.
2) Show that $Q$ and $R$ are in $\mathbb{Z}[X]$.
(Before those questions, I have shown that when two polynomials $P$,$Q$ $\in$ $\mathbb{Q}[X]$, and $PQ\in \mathbb{Z}[X]$ then $P\in\mathbb{Z}[X]$ and $Q\in\mathbb{Z}[X]$)
EDIT : I have to do this without induction, and to start with the first question.
 A: Hint:
Use induction on the degree of the dividend $A$.
The assertion is true if $\deg A<\deg B$, since $A=0\cdot B +A$.
Suppose now $\deg A=n\ge\deg B=m$, and $Q, R\in\mathbf Z[X]$ if the degree of the dividend is less than $n$. As $B$ is monic, the first step of the division algorithm consists in replacing $A$ with $A'=A-a_nX^{n-m}B$. Observe $\deg A'\le n-1$.
A: $(1)\ $ By the division algorithm in $\Bbb Q[X]\,$ we have $\, A = B\bar Q + \bar  R,\,$ $\,\bar Q,\bar R\in\Bbb Q[X].\,$ By the uniqueness of the quotient and remainder in $\,\Bbb C[X]\,$ we deduce $\,Q = \bar Q\,$ and $\,R = \bar R\,$ are both in $\,\Bbb Q[X].$
$(2)\ $ If $\,Q\,$ has a nonintegral coef then we can write $\,Q = C + Q',\,$ for $\, C\in\Bbb Z[X]\,$ and $\,Q'\,$ with nonintegral lead coef. Then $\,A = B (C+Q') + R$ $\Rightarrow\, A - BC = BQ' + R.\, $ Comparing lead coefs, using $\,B$ monic, implies that the lead coef of $\,Q'$ is integral, contradiction. So all coef's of $Q$ are integral, i.e. $\,Q\in\Bbb Z[X],\,$ hence also$\,R = A-BQ\in\Bbb Z[X]$.
