Polynomial

Definition 1. Let $R$ be a ring. The polynomial ring over $R$ is a subring of $R[[X]]$:

\[R[X] = \{p \in R[[X]] : (\exists d \in \N \, \forall n > d : p_n = 0)\}.\]

Each element of $R[X]$ is called a polynomial over $R$.

$0 \in R[X]$ is the zero polynomial.

Definition 2. Let $R$ be a ring and $p \in R[X]$ be a polynomial. The degree of $p$ is

\[\deg p = \begin{cases} -\infty, & \text{if $p = 0$}; \\ \max\{d \in \N : p_d \neq 0\}, & \text{otherwise}. \end{cases}\]

$p_0$ is called the constant term of $p$.

If $p \neq 0$, $p_{\deg p}$ is called the leading coefficient of $p$. $p$ is monic if its leading coefficient is $1$.

Proposition 1. Let $R$ be a ring. For any $p$, $q \in R[X]$,

\[\deg(p + q) \leq \max(\deg p, \deg q).\]

Proposition 2. Let $R$ be an integral domain. For any $p$, $q \in R[X]$,

\[\deg(pq) = \deg p + \deg q.\]