monic polynomial
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In
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
, a monic polynomial is a single-variable polynomial (that is, a
univariate polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
) in which the
leading coefficient In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\cdots+c_2x^2+c_1x+c_0


Univariate polynomials

If a
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtra ...

polynomial
has only one indeterminate (
univariate polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
), 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, ..., ''c''2, ''c''1 and ''c''0 are constants, the coefficients of the polynomial. Here the term ''c''''n''''x''''n'' is called the ''leading term'', and its coefficient ''c''''n'' the ''leading coefficient''; if the leading coefficient , the univariate polynomial is called monic.


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 In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation. The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', o ...
of the
polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''A'' 'x'' Actually, since the
constant polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
1 is monic, this semigroup is even a
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...
.


Partially ordered

The restriction of the
divisibility In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
relation to the set of all monic polynomials (over the given ring) is a
partial order Image:Hasse diagram of powerset of 3.svg, 250px, The Hasse diagram of the power set, set of all subsets of a three-element set , ordered by inclusion. Distinct sets on the same horizontal level are incomparable with each other. Some other pairs, su ...
, and thus makes this set to a
poset 250px, The set of all subsets of a three-element set , ordered by inclusion. Distinct sets on the same horizontal level are incomparable with each other. Some other pairs, such as and , are also incomparable. In mathematics, especially order the ...
. 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 element In the branch of abstract algebra known as ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations def ...
s other than 1.


Polynomial equation solutions

In other respects, the properties of monic polynomials and of their corresponding monic
polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s depend crucially on the coefficient ring ''A''. If ''A'' is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
, then every non-zero polynomial ''p'' has exactly one
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monic polynomial ''q'': ''p'' divided by 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. For example, 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 substituting  ''p'' = ''b''/''a''  and  ''q'' = ''c''/''a''. Thus, the equation :2x^2+3x+1 = 0 is equivalent to the monic equation :x^2+\fracx+\frac=0. The general quadratic solution formula is then the slightly more simplified form of: :x = \frac \left( -p \pm \sqrt \right).


=Integrality

= On the other hand, if the coefficient ring is not a field, there are more essential differences. For example, a monic polynomial equation with
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
coefficients cannot have
rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογι ...
solutions which are not integers. Thus, the equation :\ 2x^2+3x+1 = 0 possibly might have some rational root, which is not an integer, (and incidentally one of its roots is −1/2); while the equations :\ x^2+5x+6 = 0 and :\ x^2+7x+8 = 0 can only have integer solutions or
irrational Irrationality is cognition Cognition () refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intellectual functions and processes such as: ...
solutions. The roots of monic polynomials with integer coefficients are called
algebraic integer In algebraic number theory Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstrac ...
s. The solutions to monic polynomial equations over an
integral domain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
are important in the theory of
integral extensionIn commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' is ...
s and
integrally closed domain In commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with n ...
s, and hence for
algebraic number theory Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, ...
. 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 := \\,. 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 closureIn commutative algebra, an element ''b'' of a commutative ring In ring theory, a branch of abstract algebra, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is ca ...
of ''A'' in ''B''; or just the integral closure of ''A'', if ''B'' is the
fraction field In abstract algebra, the field of fractions of an integral domain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space ...
of ''A''; and the elements of ''C'' are said to be ''integral element, integral'' over ''A''. If here A=\mathbb (the ring of
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
s) and B=\mathbb (the field of complex numbers), 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 \mathrm(p) with elements is equal to the Necklace (combinatorics), 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 (with elements) 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 :\ 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 ''y''2 coefficient) is  2''x'' − 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''(''x''1'',...,xn'') 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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...
generated by ''x''1'',...,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" (univariate) monic polynomials. Notably, the product of monic polynomials again is monic.


See also

* Complex quadratic polynomial


Citations


References

* {{refend Polynomials