◀ ▲ ▶Branches / Algebra / Definition: Polynomial over a Ring, Degree, Variable
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).
Table of Contents
 Definition: Reduction of an Integer Polynomial Modulo a Prime Number
Mentioned in:
Algorithms: 1
Definitions: 2 3 4 5 6 7 8 9
Proofs: 10 11 12
Propositions: 13 14
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References
Bibliography
 Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013