Definition:Monic Polynomial

From ProofWiki
Jump to navigation Jump to search

Definition

Let $R$ be a commutative ring with unity.

Let $f \in R \sqbrk x$ be a polynomial in one variable over $R$.


Then $f$ is monic if and only if $f$ is nonzero and its leading coefficient is $1$.



This page has been identified as a candidate for refactoring of advanced complexity.
In particular: do something with the below
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 $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.


Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$.

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

Let $\ds f = \sum_{k \mathop = 0}^n a_k \circ x^k$ be a polynomial in $x$ over $S$.


Then $f$ is a monic polynomial if and only if its leading coefficient $a_n$ is $1_R$.


Polynomial Form

Let $R$ be a commutative ring with unity $1_R$.

Let $f = a_0 + a_1 X + \cdots + a_{r-1} X^{r-1} + a_r X^r$ be a polynomial from in the single indeterminate $X$ over $R$.


Then $f$ is monic if the leading coefficient of $f$ is $1_R$.


Sequence

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


Then $f$ is a monic polynomial if and only if its leading coefficient $a_n$ is $1$.


Also see

  • Results about monic polynomials can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Monic_Polynomial&oldid=733762"