Definition:Degree of Polynomial

General Ring
Let $$R$$ be a ring with unity.

Let $$R \left[{x}\right]$$ be the polynomial over $R$ in $x$.

Let $$z = r_0 + r_1 x + \cdots + r_n x^n$$ be an element of $$R \left[{x}\right]$$.

Then the degree of $$z$$ is $$n$$, and can be denoted $$\partial z = n$$.

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

Let $$\left({D, +, \circ}\right)$$ be an integral domain such that $$D$$ is a subring of $$R$$.

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

Let $$D \left[{X}\right]$$ be the ring of polynomial forms in $$X$$ over $$D$$.

Let $$f$$ be a non-zero element of $$D \left[{X}\right]$$.

By Unique Representation in Polynomial Forms‎, there is one way of expressing $$f$$ as a polynomial:

$$f \in D \left[{X}\right]: f = \sum_{k=0}^n {a_k \circ X^k}$$

In particular, the coefficients $$a_0, a_1, \ldots, a_n$$ are uniquely determined by $$f$$.

The number $$n$$ is called the degree of $$f$$, or its order, and can be denoted $$\partial f$$.

Degree Zero
A polynomial of degree zero is a non-zero element of $$D$$, that is, a constant.

Null Polynomial
The null polynomial $$0_R \in D \left[{X}\right]$$ does not have a degree.

Alternative names
The degree of a polynomial is also referred to as its order.