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

### Related definitions:

• 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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### References

#### Bibliography

1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013