Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Monic polynomial
Summary
In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one that can be written as
with
Monic polynomials are widely used in algebra and number theory, since they produce many simplifications and they avoid divisions and denominators. Here are some examples.
Every polynomial is associated to a unique monic polynomial. In particular, the unique factorization property of polynomials can be stated as: Every polynomial can be uniquely factorized as the product of its leading coefficient and a product of monic irreducible polynomials.
Vieta's formulas are simpler in the case of monic polynomials: The ith elementary symmetric function of the roots of a monic polynomial of degree n equals where is the coefficient of the (n−i)th power of the indeterminate.
Euclidean division of a polynomial by a monic polynomial does not introduce divisions of coefficients. Therefore, it is defined for polynomials with coefficients in a commutative ring.
Algebraic integers are defined as the roots of monic polynomials with integer coefficients.
Every nonzero univariate polynomial (polynomial with a single indeterminate) can be written
where are the coefficients of the polynomial, and the leading coefficient is not zero. By definition, such a polynomial is monic if
A product of polynomials is monic if and only if all factors are monic.
The "if" condition 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 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 induces a partial order on monic polynomials. This results almost immediately from the preceding properties.
Let be a polynomial equation, where P is a univariate polynomial of degree n.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.