Definition:Degree of Polynomial/Zero
Jump to navigation
Jump to search
This page has been identified as a candidate for refactoring. In particular:
should be a theorem, not a definition
should be a theorem, not a definition
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
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 $S$.
A polynomial $f \in S \sqbrk x$ in $x$ over $S$ is of degree zero if and only if $x$ is a non-zero element of $S$, that is, a constant polynomial.
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 64$. Polynomial rings over an integral domain: Remarks $\text{(a)}$