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 [1]
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 monic polynomials is monic. A product of polynomials is monic if and only if the product of the leading coefficients of the factors equals 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 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. If one divides all coefficients of P by its leading coefficient one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial.
For example, the equation
is equivalent to the monic equation
When the coefficients are unspecified, or belong to a field where division does not result into fractions (such as or a finite field), this reduction to monic equations may provide simplification. On the other hand, as shown by the previous example, when the coefficients are explicit integers, the associated monic polynomial is generally more complicated. Therefore, primitive polynomials are often used instead of monic polynomials when dealing with integer coefficients.
Monic polynomial equations are at the basis of the theory of algebraic integers, and, more generally of integral elements.
Let R be a subring of a field F; this implies that R is an integral domain. An element a of F is integral over R if it is a root of a monic polynomial with coefficients in R.
A complex number that is integral over the integers is called an algebraic integer. This terminology is motivated by the fact that the integers are exactly the rational numbers that are also algebraic integers. This results from the rational root theorem, which asserts that, if the rational number is a root of a polynomial with integer coefficients, then q is a divisor of the leading coefficient; so, if the polynomial is monic, then and the number is an integer. Conversely, an integer p is a root of the monic polynomial
It can be proved that, if two elements of a field F are integral over a subring R of F, then the sum and the product of these elements are also integral over R. It follows that the elements of F that are integral over R form a ring, called the integral closure of R in K. An integral domain that equals its integral closure in its field of fractions is called an integrally closed domain.
These concepts are fundamental in algebraic number theory. For example, many of the numerous wrong proofs of the Fermat's Last Theorem that have been written during more than three centuries were wrong because the authors supposed wrongly that the algebraic integers in an algebraic number field have unique factorization.
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 one variable with coefficients being polynomials in the other variables. Being monic depends thus on the choice of one "main" variable. For example, the polynomial
is monic, if considered as a polynomial in x with coefficients that are polynomials in y:
but it is not monic when considered as a polynomial in y with coefficients polynomial in x:
In the context of Gröbner bases, a monomial order is generally fixed. In this case, a polynomial may be said to be monic, if it has 1 as its leading coefficient (for the monomial order).
For every definition, a product of monic polynomials is monic, and, if the coefficients belong to a field, every polynomial is associated to exactly one monic polynomial.
In mathematics, Bézout's identity, named after Étienne Bézout who proved it for polynomials, is the following theorem:
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.
In mathematics, a polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1.
In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or an expression. It may be a number (dimensionless), in which case it is known as a numerical factor. It may also be a constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any expression. When the combination of variables and constants is not necessarily involved in a product, it may be called a parameter.
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.
In mathematics, a unique factorization domain (UFD) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain in which every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units.
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x2 – 4.
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial whose coefficients are integers. The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as if it is considered as a polynomial with real coefficients. One says that the polynomial x2 − 2 is irreducible over the integers but not over the reals.
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.
In mathematics, an algebraic equation or polynomial equation is an equation of the form , where P is a polynomial with coefficients in some field, often the field of the rational numbers. For example, is an algebraic equation with integer coefficients and
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root, or, equivalently, a common factor. In some older texts, the resultant is also called the eliminant.
In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems.
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words,
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts.
In algebra, the greatest common divisor of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers.
In algebra, the content of a nonzero polynomial with integer coefficients is the greatest common divisor of its coefficients. The primitive part of such a polynomial is the quotient of the polynomial by its content. Thus a polynomial is the product of its primitive part and its content, and this factorization is unique up to the multiplication of the content by a unit of the ring of the coefficients.
In commutative algebra, an integrally closed domainA is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if x is an element of the field of fractions of A which is a root of a monic polynomial with coefficients in A, then x is itself an element of A. Many well-studied domains are integrally closed, as shown by the following chain of class inclusions:
