We consider $P = f _ { 0 } + f _ { 1 } X + \cdots + f _ { n } X ^ { n } \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$, with $\lambda = 0$, $f _ { 0 } ( 0 ) = \cdots = f _ { d - 1 } ( 0 ) = 0$ and $f _ { d }$ the constant function equal to 1. Let $F \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { d } [ X ] \right)$ be monic and such that $F _ { \mid t = 0 } = X ^ { d }$. Let $R$ be the remainder of the Euclidean division of $P$ by $F$. Show that $F + R$ is monic of degree $d$ and that $( F + R ) _ { \mid t = 0 } = X ^ { d }$.
We consider $P = f _ { 0 } + f _ { 1 } X + \cdots + f _ { n } X ^ { n } \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$, with $\lambda = 0$, $f _ { 0 } ( 0 ) = \cdots = f _ { d - 1 } ( 0 ) = 0$ and $f _ { d }$ the constant function equal to 1.
Let $F \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { d } [ X ] \right)$ be monic and such that $F _ { \mid t = 0 } = X ^ { d }$. Let $R$ be the remainder of the Euclidean division of $P$ by $F$. Show that $F + R$ is monic of degree $d$ and that $( F + R ) _ { \mid t = 0 } = X ^ { d }$.