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 $1$.
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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): monic polynomial
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): monic polynomial
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): monic polynomial