Monic polynomial

In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form

$x^{n}+c_{n-1}x^{n-1}+\cdots +c_{2}x^{2}+c_{1}x+c_{0}$

Univariate polynomials

If a polynomial has only one indeterminate (univariate polynomial), then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). A univariate polynomial in x of degree n then takes the general form displayed above, where

c_n ≠ 0, c_n−1, ..., c₂, c₁ and c₀

are constants, the coefficients of the polynomial.

Here the term c_nxⁿ is called the leading term, and its coefficient c_n the leading coefficient; if the leading coefficient is 1, the univariate polynomial is called monic.

Examples

• Complex quadratic polynomials

Properties

Multiplicatively closed

The set of all monic polynomials (over a given (unitary) ring A and for a given variable x) is closed under multiplication, since the product of the leading terms of two monic polynomials is the leading term of their product. Thus, the monic polynomials form a multiplicative semigroup of the polynomial ring A[x]. Actually, since the constant polynomial 1 is monic, this semigroup is even a monoid.

Partially ordered

The restriction of the divisibility relation to the set of all monic polynomials (over the given ring) is a partial order, and thus makes this set to a poset. The reason is that if p(x) divides q(x) and q(x) divides p(x) for two monic polynomials p and q, then p and q must be equal. The corresponding property is not true for polynomials in general, if the ring contains invertible elements other than 1.

Polynomial equation solutions

In other respects, the properties of monic polynomials and of their corresponding monic polynomial equations depend crucially on the coefficient ring A. If A is a field, then every non-zero polynomial p has exactly one associated monic polynomial q; actually, q is p divided with its leading coefficient. In this manner, then, any non-trivial polynomial equation p(x) = 0 may be replaced by an equivalent monic equation q(x) = 0. E.g., the general real second degree equation

$\ ax^{2}+bx+c=0$ (where $a\neq 0$)

may be replaced by

$\ x^{2}+px+q=0$,

by putting  p = b/a  and  q = c/a. Thus, the equation

$2x^{2}+3x+1=0$

is equivalent to the monic equation

$x^{2}+{\frac {3}{2}}x+{\frac {1}{2}}=0.$

The general quadratic solution formula is then the slightly more simplified form of:

$x={\frac {1}{2}}\left(-p\pm {\sqrt {p^{2}-4q}}\right).$

Integrality

On the other hand, if the coefficient ring is not a field, there are more essential differences. E.g., a monic polynomial equation with integer coefficients cannot have other rational solutions than integer solutions. Thus, the equation

$\ 2x^{2}+3x+1=0$

possibly might have some rational root, which is not an integer, (and incidentally it does have inter alia the root −1/2); while the equations

$\ x^{2}+5x+6=0$

and

$\ x^{2}+7x+8=0$

only may have integer solutions or irrational solutions.

The roots of monic polynomial with integer coefficients are called algebraic integers.

The solutions to monic polynomial equations over an integral domain are important in the theory of integral extensions and integrally closed domains, and hence for algebraic number theory. In general, assume that A is an integral domain, and also a subring of the integral domain B. Consider the subset C of B, consisting of those B elements, which satisfy monic polynomial equations over A:

$C:=\{b\in B:\exists \,p(x)\in A[x]\,,{\hbox{ which is monic and such that }}p(b)=0\}\,.$

The set C contains A, since any a ∈ A satisfies the equation x − a = 0. Moreover, it is possible to prove that C is closed under addition and multiplication. Thus, C is a subring of B. The ring C is called the integral closure of A in B; or just the integral closure of A, if B is the fraction field of A; and the elements of C are said to be integral over A. If here $A=\mathbb {Z}$ (the ring of integers) and $B=\mathbb {C}$ (the field of complex numbers), then C is the ring of algebraic integers.

Irreduciblity

If p is a prime number, the number of monic irreducible polynomials of degree n over a finite field $GF(p)$ with p elements is equal to the necklace counting function $N_{p}(n)$.

If one removes the constraint of being monic, this number becomes $(p-1)N_{p}(n)$.

The total number of roots of these monic irreducible polynomials is $nN_{p}(n)$. This is the number of elements of the field $GF(p^{n})$ (with $p^{n}$ elements) that do not belong to any smaller field.

For p = 2, such polynomials are commonly used to generate pseudorandom binary sequences.

Multivariate polynomials

Ordinarily, the term monic is not employed for polynomials of several variables. However, a polynomial in several variables may be regarded as a polynomial in only "the last" variable, but with coefficients being polynomials in the others. This may be done in several ways, depending on which one of the variables is chosen as "the last one". E.g., the real polynomial

$\ p(x,y)=2xy^{2}+x^{2}-y^{2}+3x+5y-8$

is monic, considered as an element in R[y][x], i.e., as a univariate polynomial in the variable x, with coefficients which themselves are univariate polynomials in y:

$p(x,y)=1\cdot x^{2}+(2y^{2}+3)\cdot x+(-y^{2}+5y-8)$;

but p(x,y) is not monic as an element in R[x][y], since then the highest degree coefficient (i.e., the y² coefficient) is  2x − 1.

There is an alternative convention, which may be useful e.g. in Gröbner basis contexts: a polynomial is called monic, if its leading coefficient (as a multivariate polynomial) is 1. In other words, assume that p = p(x₁,...,x_n) is a non-zero polynomial in n variables, and that there is a given monomial order on the set of all ("monic") monomials in these variables, i.e., a total order of the free commutative monoid generated by x₁,...,x_n, with the unit as lowest element, and respecting multiplication. In that case, this order defines a highest non-vanishing term in p, and p may be called monic, if that term has coefficient one.

"Monic multivariate polynomials" according to either definition share some properties with the "ordinary" (univariate) monic polynomials. Notably, the product of monic polynomials again is monic.

References

• Pinter, Charles C. (2010) [Unabridged republication of the 1990 second edition of the work originally published in 1982 by the McGraw–Hill Publishing Company]. A Book of Abstract Algebra. Dover. ISBN 978-0486474175.