Definition: Polynomial over a Ring, Degree, Variable

A polynomial over the commutative ring $R$ is a term

\[p:=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n}\]

with $a_{i}\in R,\,i=0,\ldots ,n,\,n\in \mathbb {N} $.

The symbol $x$ is called an indeterminate or variable.
The maximal natural number $n$ with $a_n\neq 0$ is called the degree of the polynomial, and denoted by $\deg(p).$
By convention, the degree of the zero polynomial, i.e. the polynomial $p$ with $p=0$ is set to $-\infty.$
A polynomial $p$ with $\deg(p)=-\infty$ or $\deg(p)=0$ is called a constant polynomial.
A polynomial $p$ with $\deg(p)=1$ is called a linear polynomial.
A polyonmial $p$ with $\deg(p)=n$ and $a_n=1$ is called normed or monic).

Algorithms: 1
Definitions: 2 3 4 5 6 7 8 9
Proofs: 10 11 12
Propositions: 13 14

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