If a polynomial has only one indeterminate, then the terms are usually written either from highest degree to lowest degree or from lowest degree to highest degree. A univariate polynomial in x of degree n then takes the general form displayed above, where are constants, the coefficients of the polynomial. Here the term cnxn is called the leading term, and its coefficient cn the leading coefficient; if the leading coefficient, the univariate polynomial is called monic.
Examples
Complex quadratic polynomials
Properties
Multiplicatively closed
The set of all monic polynomials 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 ringA. 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 is a partial order, and thus makes this set to a poset. The reason is that if p divides q and q divides p 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.
In other respects, the properties of monic polynomials and of their corresponding monic polynomial equations depend crucially on the coefficient ringA. If A is a field, then every non-zero polynomial p has exactly oneassociated monic polynomial q; actually, q is p divided with its leading coefficient. In this manner, then, any non-trivial polynomial equationp = 0 may be replaced by an equivalent monic equation q = 0. E.g., the general real second degree equation may be replaced by by putting p = b/a and q = c/a. Thus, the equation is equivalent to the monic equation The general quadratic solution formula is then the slightly more simplified form of:
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 possibly might have some rational root, which is not an integer, ; while the equations and 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 ofintegral 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: 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 and , then C is the ring of algebraic integers.
Irreduciblity
If is a prime number, the number of monic irreducible polynomials of degree over a finite field with elements is equal to the necklace counting function. If one removes the constraint of being monic, this number becomes. The total number of roots of these monic irreducible polynomials is. This is the number of elements of the field that do not belong to any smaller field. For, 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 is monic, considered as an element in R, i.e., as a univariate polynomial in the variable x, with coefficients which themselves are univariate polynomials in y: but p is not monic as an element in R, since then the highest degree 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 is 1. In other words, assume that p = p is a non-zero polynomial in n variables, and that there is a given monomial order on the set of all monomials in these variables, i.e., a total order of the free commutative monoid generated by x1,...,xn, 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" monic polynomials. Notably, the product of monic polynomials again is monic.
