# Definition:Monic Polynomial

## 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 belowUntil this has been finished, please leave
`{{Refactor}}` in the code.
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

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 64$. Polynomial rings over an integral domain - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**monic polynomial** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**monic polynomial** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**monic polynomial**