Definition:Degree of Polynomial/Null Polynomial

From ProofWiki
Jump to navigation Jump to search

Definition

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

Let $\struct {S, +, \circ}$ be a subring of $R$.

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


The null polynomial $0_R \in S \sqbrk x$ 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.


Also see

  • Results about the degree of a null polynomial can be found here.


Sources