Definition:Degree of Polynomial/Null Polynomial

Definition

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

Let $\left({S, +, \circ}\right)$ be a subring of $R$.

For arbitrary $x \in R$, let $S \left[{x}\right]$ be the set $S \left[{x}\right]$ be the set of polynomials in $x$ over $S$.

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

Integral Domain

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

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

For arbitrary $x \in R$, let $D \sqbrk x$ be the ring of polynomials in $x$ over $D$.

The null polynomial $0_R \in D \sqbrk x$ does not have a degree.

Sequence

Let $f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$ be a polynomial over a field $F$.

Let $0_F$ be the zero of $F$.

Let $a_0 = a_1 = a_2 = \ldots = 0_F$.

Then $f$ is a null polynomial over $F$.

Also defined as

Some sources assign the value of $-\infty$ to the degree of the null polynomial.