Definition:Monic Polynomial

From ProofWiki
Jump to: navigation, search

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