Monic polynomial

Inalgebra,amonic polynomialis a non-zerounivariate polynomial(that is, a polynomial in a single variable) in which theleading 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

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

with$n\geq 0.$

Uses

In this section, "polynomial" is used as a shorthand forunivariate polynomial,and, unless explicitly stated the coefficients of the polynomials belong to a fixedfield.

Monic polynomials are widely used inalgebraandnumber theory,since they produce many simplifications and they avoid divisions and denominators. Here are some examples.

Every polynomial isassociatedto a unique monic polynomial. In particular, theunique factorizationproperty of polynomials can be stated as:Every polynomial can be uniquely factorized as the product of itsleading coefficientand a product of monicirreducible polynomials.

Vieta's formulasare simpler in the case of monic polynomials:Theithelementary symmetric functionof therootsof a monic polynomial of degreenequals$(-1)^{i}c_{n-i},$where$c_{n-i}$is the coefficient of the(n−i)th power of theindeterminate.

Euclidean divisionof a polynomial by a monic polynomial does not introduce divisions of coefficients. Therefore, it is defined for polynomials with coefficients in acommutative ring.

Algebraic integersare defined as the roots of monic polynomials with integer coefficients.

Properties

Every nonzerounivariate polynomial(polynomialwith a singleindeterminate) can be written

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

where$c_{n},\ldots,c_{0}$are the coefficients of the polynomial, and theleading coefficient$c_{n}$is not zero. By definition, such a polynomial ismonicif$c_{n}=1.$

A product of monic polynomials is monic. A product of polynomials is monicif and only ifthe product of the leading coefficients of the factors equals1.

This implies that, the monic polynomials in a univariatepolynomial ringover acommutative ringform amonoidunder polynomial multiplication.

Two monic polynomials areassociatedif 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.

Divisibilityinduces apartial orderon monic polynomials. This results almost immediately from the preceding properties.

Polynomial equations

Let$P(x)$be apolynomial equation,wherePis aunivariate polynomialof degreen.If one divides all coefficients ofPby itsleading coefficient$c_{n},$one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial.

For example, the equation

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

is equivalent to the monic equation

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

When the coefficients are unspecified, or belong to afieldwhere division does not result into fractions (such as$\mathbb {R},\mathbb {C},$or afinite 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 polynomialsare often used instead of monic polynomials when dealing with integer coefficients.

Integral elements

Monic polynomial equations are at the basis of the theory ofalgebraic integers,and, more generally ofintegral elements.

LetRbe a subring of afieldF;this implies thatRis anintegral domain.An elementaofFisintegraloverRif it is arootof a monic polynomial with coefficients inR.

Acomplex numberthat is integral over the integers is called analgebraic integer.This terminology is motivated by the fact that the integers are exactly therational numbersthat are also algebraic integers. This results from therational root theorem,which asserts that, if the rational number${\textstyle {\frac {p}{q}}}$is a root of a polynomial with integer coefficients, thenqis a divisor of the leading coefficient; so, if the polynomial is monic, then$q=\pm 1,$and the number is an integer. Conversely, an integerpis a root of the monic polynomial$x-a.$

It can be proved that, if two elements of a fieldFare integral over a subringRofF,then the sum and the product of these elements are also integral overR.It follows that the elements ofFthat are integral overRform a ring, called theintegral closureofRinK.An integral domain that equals its integral closure in itsfield of fractionsis called anintegrally closed domain.

These concepts are fundamental inalgebraic number theory.For example, many of the numerous wrong proofs of theFermat's Last Theoremthat have been written during more than three centuries were wrong because the authors supposed wrongly that the algebraic integers in analgebraic number fieldhaveunique factorization.

Multivariate polynomials

Ordinarily, the termmonicis 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. Beingmonicdepends thus on the choice of one "main" variable. For example, the polynomial

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

is monic, if considered as a polynomial inxwith coefficients that are polynomials iny:

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

but it is not monic when considered as a polynomial inywith coefficients polynomial inx:

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

In the context ofGröbner bases,amonomial orderis 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 afield,every polynomial isassociatedto exactly one monic polynomial.

References

• Fraleigh, John B. (2003).A First Course in Abstract Algebra(7th ed.).Pearson Education.ISBN9780201763904.