Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Monic polynomial
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== Properties == Every nonzero [[univariate polynomial]] ([[polynomial]] with a single [[indeterminate (variable)|indeterminate]]) can be written :<math>c_nx^n + c_{n-1}x^{n-1}+ \cdots c_1x +c_0,</math> where <math>c_n,\ldots,c_0</math> are the coefficients of the polynomial, and the [[leading coefficient]] <math>c_n</math> is not zero. By definition, such a polynomial is ''monic'' if <math>c_n=1.</math> A product of monic polynomials is monic. A product of polynomials is monic [[if and only if]] the product of the leading coefficients of the factors equals {{math|1}}. This implies that, the monic polynomials in a univariate [[polynomial ring]] over a [[commutative ring]] form a [[monoid]] under polynomial multiplication. Two monic polynomials are [[associated element|associated]] if and only if they are equal, since the multiplication of a polynomial by a nonzero constant produces a polynomial with this constant as its leading coefficient. [[Divisibility (ring theory)|Divisibility]] induces a [[partial order]] on monic polynomials. This results almost immediately from the preceding properties.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)