Definition:Degree of Polynomial/Integral Domain

Definition
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity whose zero is $0_R$.

Let $\left({D, +, \circ}\right)$ be an integral subdomain of $R$.

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

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

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

The degree of $f$ can be denoted $\deg \left({f}\right)$ or $\partial f$.

Also known as
The degree of a polynomial is also referred to by some sources as its order.