Why is the restriction of being "Monic " is imposed on the divisor element?

Question

I'm reading the following proof,upto now i'd read division algorithm only for $\mathbb F[x],$where $\mathbb F$ is field.In this proof i first time encountered with the notion of division algorithm in $\mathbb Z[x]$,where $\mathbb Z$ is a set of integers(integral domain).

I've two queries regarding the following proof-

• Why is the restriction of being "Monic " is imposed on the divisor element?
• Where does the division algorithm fail if above restriction is removed?

Here is the main source https://brilliant.org/wiki/cyclotomic-polynomials/

Answer 1

The divisor has to be monic, because otherwise the result is not necessarily true. It is easy to find two polynomials $f, g$ in ${\mathbb Z}[x]$ such that $g | f$ in ${\mathbb C}[x]$ but whose quotient $f/g$ is not in ${\mathbb Z}[x]$. For instance, as @ancientmathematician mentions in a comment, $f = 1$ and $g = 2$.

The normal division algorithm cancels the highest term of $f$ by subtracting a suitable multiple of $g$ (and then repeating this until the degree of $f$ drops below the degree of $g$). This suitable multiple is $(\text{lc}(f) / \text{lc}(g)) g$. If $\text{lc}(g) = 1$ (or $-1$), this is guaranteed to be in ${\mathbb Z}[x]$.