Polynomial ring/Field/Polynomial function/Introduction/Section

Definition

The polynomial ring over a field ${}K$ consists of all polynomials

${}P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,,$

where ${}a_{i}\in K$, ${}n\in \mathbb {N}$. It is endowed with componentwise addition and a multiplication, which arises by distributive continuation of the rule

${}X^{n}\cdot X^{m}:=X^{n+m}\,.$

A polynomial

${}P=\sum _{i=0}^{n}a_{i}X^{i}=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,$

is, formally seen, nothing but the tuple ${}(a_{0},a_{1},\ldots ,a_{n})$, these numbers are called the coefficient of the polynomial. The polynomials are equal if and only if all their coefficients coincide. The letter ${}X$ is called the variable of the polynomial ring. In this context, the field ${}K$ is called the base field of the polynomial ring. Due to the componentwise definition of the addition, we have immediately a commutative group, with the zero polynomial (where all coefficients are ${}0$) as neutral element. The polynomials with ${}a_{i}=0$ for all ${}i\geq 1$ are called constant polynomials, they are simply written as ${}a_{0}$.

The way a polynomial is written suggests how the multiplication shall work, the product ${}X^{n}\cdot X^{m}$ is given by the addition of the exponents, thus ${}X^{n}\cdot X^{m}:=X^{n+m}$. For arbitrary polynomials, the multiplication arises from this simple multiplication rule by distributive continuation according to the law to multiply "everything with everything“. Explicitly, the multiplication is given by the following rule gegeben:

${\left(\sum _{i=0}^{n}a_{i}X^{i}\right)}\cdot {\left(\sum _{j=0}^{m}a_{j}X^{j}\right)}=\sum _{k=0}^{n+m}c_{k}X^{k}{\text{ where }}c_{k}=\sum _{r=0}^{k}a_{r}b_{k-r}.$

The multiplication is associative, commutative, distributive, and the constant polynomial ${}1$ is its neutral element, see exercise. Altogether, we have a commutative ring.

Definition

The degree of a nonzero polynomial

${}P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,$

with ${}a_{n}\neq 0$

is ${}n$.

We do not define a degree for the zero polynomial. The coefficient ${}a_{n}$, where ${}n$ is the degree of the polynomial, is called the leading coefficient of the polynomial. The term ${}a_{n}X^{n}$ is called leading term. A polynomial with leading coefficient ${}1$ is called normed.

The graph of a polynomial function from ${}\mathbb {R}$ to ${}\mathbb {R}$ of degree ${}5$.

We can plug in (or insert or evaluate at) an element ${}a\in K$, into a polynomial ${}P\in K[X]$, by replacing the variable ${}X$ everywhere by ${}a$. This gives a mapping

$K\longrightarrow K,a\longmapsto P(a),$

which we call the polynomial function defined by the polynomial. In general, this mapping is not linear, only the polynomials of the form ${}P=a_{1}X$ are linear.