# Definition:Monic Polynomial

## Definition

Let $R$ be a commutative ring with unity.

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

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

Let $\left({R, +, \circ}\right)$ be a ring with unity whose unity is $1_R$.

Let $\left({S, +, \circ}\right)$ be a subring of $R$.

Let $\displaystyle f = \sum_{k \mathop = 0}^n a_k \circ x^k$ 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 $f = \left \langle {a_k}\right \rangle = \left({a_0, a_1, a_2, \ldots}\right)$ be a polynomial over a field $F$.

Then $f$ is a **monic polynomial** iff its leading coefficient $a_n$ is $1$.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 64$