Editing Monic polynomial

{{Short description|Polynomial with 1 as leading coefficient}}
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{{sfn|Fraleigh|2003|p=432|loc=Under the Prop. 11.29}}
:<math>x^n+c_{n-1}x^{n-1}+\cdots+c_2x^2+c_1x+c_0,</math>
with <math>n \geq 0.</math>

== Uses ==
{{hatnote|In this section, "polynomial" is used as a shorthand for [[univariate polynomial]], and, unless explicitly stated the coefficients of the polynomials belong to a fixed [[field (mathematics)|field]].}}

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 elements|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 polynomial]]s.''

[[Vieta's formulas]] are simpler in the case of monic polynomials: ''The {{mvar|i}}th [[elementary symmetric function]] of the [[polynomial root|roots]] of a monic polynomial of degree {{math|n}} equals <math>(-1)^ic_{n-i},</math> where <math>c_{n-i}</math> is the coefficient of the {{math|(n−i)}}th power of the [[indeterminate (variable)|indeterminate]].''

[[Euclidean division of polynomials|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 integer]]s are defined as the roots of monic polynomials with integer coefficients.

== Properties ==
Every nonzero [[univariate polynomial]] ([[polynomial]] with a single [[indeterminate (variable)|indeterminate]]) can be written  
:<math>c_nx^n + c_{n-1}x^{n-1}+ \cdots c_1x +c_0,</math>
where <math>c_n,\ldots,c_0</math> are the coefficients of the polynomial, and the [[leading coefficient]] <math>c_n</math> is not zero. By definition, such a polynomial is ''monic'' if <math>c_n=1.</math>

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 {{math|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 element|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 (ring theory)|Divisibility]] induces a [[partial order]] on monic polynomials. This results almost immediately from the preceding properties.

==Polynomial equations==

Let <math>P(x)</math> be a [[polynomial equation]], where {{mvar|P}} is a [[univariate polynomial]] of degree {{mvar|n}}. If one divides all coefficients of {{mvar|P}} by its [[leading coefficient]] <math>c_n,</math> one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial.

For example, the equation
:<math>2x^2+3x+1 = 0</math>
is equivalent to the monic equation
:<math>x^2+\frac{3}{2}x+\frac{1}{2}=0.</math>

When the coefficients are unspecified, or belong to a [[field (mathematics)|field]] where division does not result into fractions (such as <math>\R, \Complex,</math> 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 polynomial (ring theory)|primitive polynomial]]s are often used instead of monic polynomials when dealing with integer coefficients.

==Integral elements==
Monic polynomial equations are at the basis of the theory of [[algebraic integer]]s, and, more generally of [[integral element]]s.

Let {{mvar|R}} be a subring of a [[field (mathematics)|field]] {{mvar|F}}; this implies that {{mvar|R}} is an [[integral domain]]. An element {{mvar|a}} of {{mvar|F}} is ''integral'' over {{mvar|R}} if it is a [[polynomial root|root]] of a monic polynomial with coefficients in {{mvar|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 number]]s that are also algebraic integers. This results from the [[rational root theorem]], which asserts that, if the rational number <math display =inline>\frac pq</math> is a root of a polynomial with integer coefficients, then {{mvar|q}} is a divisor of the leading coefficient; so, if the polynomial is monic, then <math>q=\pm 1,</math> and the number is an integer. Conversely, an integer {{mvar|p}} is a root of the monic polynomial <math>x-a.</math>

It can be proved that, if two elements of a field {{mvar|F}} are integral over a subring   {{mvar|R}} of {{mvar|F}}, then the sum and the product of these elements are also integral over {{mvar|R}}. It follows that the elements of {{mvar|F}} that are integral over {{mvar|R}} form a ring, called the [[integral closure]] of {{mvar|R}} in {{mvar|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]].

== 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 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
:<math>p(x,y) = 2xy^2+x^2-y^2+3x+5y-8</math>
is monic, if considered as a polynomial in {{mvar|x}} with coefficients that are  polynomials in {{mvar|y}}:
:<math>p(x,y) = x^2 + (2y^2+3) \, x + (-y^2+5y-8);</math>
but it is not monic when considered as a polynomial in {{mvar|y}} with coefficients polynomial in {{mvar|x}}:
:<math>p(x,y)=(2x-1)\,y^2+5y +(x^2+3x-8).</math>

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 (mathematics)|field]], every polynomial is [[associated element|associated]] to exactly one monic polynomial.

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
* {{cite book | last=Fraleigh | first=John B. | title = A First Course in Abstract Algebra | year=2003 | edition=7th |publisher=[[Pearson Education]] | url=https://www.pearson.com/us/higher-education/program/Fraleigh-First-Course-in-Abstract-Algebra-A-7th-Edition/PGM44169.html | isbn=9780201763904}}
{{refend}}
[[Category:Polynomials]]