Suppose a and b are polynomials with integer coefficients and b has a leading coefficient 1. Prove that the quotient and remainder of division a and b in $\mathbb{Q}$[X] also belong to $\mathbb{Z}$[X].

While i understand when b has an leading coefficient 1. It can atleast divide the highest degree of a as long as the degree of a is higer than b. Meaning the quotient q will be an integer. But for the remainder r i cannot fully comprehend that it also will be an integer. Any hints or assumptions I should take?

  • $\begingroup$ Just look at how division is performed in $\mathbb{Q}[X]$ and you'll realize that you never go outside $\mathbb{Z}$ if both $a$ and $b$ are in $\mathbb{Z}[X]$ and $b$ is monic. $\endgroup$ – egreg Oct 14 '18 at 22:07

Let $ a = q\, b + r.\,$ If $\,q\,$ has a coef $\not\in\Bbb Z\,$ write $\,q = p + q',\,$ for $\, p\in\Bbb Z[X]\,$ and $\,q'\,$ with lead coef $\not\in\Bbb Z.\,$ $a = b (p+q') + r$ $\Rightarrow\, a - bp = bq' + r.\, $ so comparing lead coefs using $\,b$ monic & $\deg r < \deg b\,$ yields $\,q'$ has lead coef $\in\Bbb Z,\,$ contradiction. Thus $\,q\in\Bbb Z[x],\,$ so $\,r = a\!-\!qb\in\Bbb Z[X]$.

Alternatively it is the special case $c=1$ of the nonmonic division algorithm, which implies that $\ c^k a = q\,b + r\,$ for $\,q,r\in\Bbb Z[X],\ c$ = lead coef of $\,b.$


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