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.
Sources
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 3.2$: Polynomial rings: Definition $3.6$