📂Abstract Algebra

Ring of Polynomial Functions

Definition¹

$$ f(x) : = \sum_{k=0}^{n} a_{k} x^{k} = a_{0} + a_{1} x + \cdots + a_{n} x^{n} $$ A polynomial $f(x)$ over a ring $R$ is defined as above.

1. The $a_{i} \in R$ are called the coefficients of $f(x)$.
2. If $n < \infty$, then $n$ is called the degree of $f(x)$.
3. $R[x]$ is the set of finite polynomial functions whose coefficients are elements of $R$. $$ R[x] := \left\{ a_{0} + a_{1} x + \cdots + a_{n} x^{n} \ | \ a_{0}, \cdots , a_{n} \in R \right\} $$
4. $R[[x]]$ is the set of infinite polynomial functions whose coefficients are elements of $R$. $$ R[[x]] := \left\{ a_{0} + a_{1} x + \cdots + a_{n} x^{n} + \cdots \ | \ a_{0}, \cdots , a_{n} , \cdots \in R \right\} $$

Explanation

After a long, winding detour, we have finally returned to the ‘algebra’ we learned in middle and high school. The reason for defining polynomial functions anew is to view and handle a polynomial ’expression’ as an element of a group, ring, or field.

To this end, one should know the following important theorems. They may not seem like much, but they are facts guaranteeing that the ring of polynomials preserves the useful properties of the original ring.

Theorem

• [1]: If $R$ is a ring, then $R[x]$ is also a ring with respect to the addition and multiplication of polynomial functions.
• [2]: If $R$ is a commutative ring, then $R[x]$ is also a commutative ring.
• [3]: If $R$ has unity $1 \ne 0$, then $R[x]$ also has unity $1 \ne 0$.

If these theorems hold for $R[x]$, then they hold for $R[[x]]$ as well.