Definition:Degree of Polynomial/Integral Domain

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Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$.

Let $\struct {D, +, \circ}$ be an integral subdomain of $R$.

Let $X \in R$ be transcendental over $D$.

Let $\ds f = \sum_{j \mathop = 0}^n \paren {r_j \circ X^j} = r_0 + r_1 X + \cdots + r_n X^n$ be a polynomial over $D$ in $X$ such that $r_n \ne 0$.

Then the degree of $f$ is $n$.

The degree of $f$ is denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ by $\map \deg f$.

Also known as

The degree of a polynomial $f$ is also sometimes called the order of $f$.

Some sources denote $\map \deg f$ by $\partial f$ or $\map \partial f$.