📂Abstract Algebra

Polynomial Rings

Definition ¹

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

1. $a_{i} \in R$ 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 all polynomials with coefficients in $R$. $$R[x] := \left\{ a_{0} + a_{1} x + \cdots + a_{n} x^{n} \ | \ a_{0}, \cdots , a_{n} \in R \right\}$$
4. $R[[x]]$ is the set of all formal power series with coefficients in $R$. $$R[[x]] := \left\{ a_{0} + a_{1} x + \cdots + a_{n} x^{n} + \cdots \ | \ a_{0}, \cdots , a_{n} , \cdots \in R \right\}$$

Explanation

After a long detour, we are back to the ‘algebra’ learned in middle and high school. The reason for redefining polynomials is to treat polynomial ’equations’ as elements of groups, rings, and fields.

It is essential to know the following important theorems. They may not seem like much but they guarantee that the ring of polynomials preserves the useful properties of the original ring.

Theorems

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

These theorems hold for $R[x]$ also apply to $R[[x]]$.