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In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (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 consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
has only one indeterminate ( univariate polynomial), 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 of the polynomial ring ''A'' 'x'' Actually, since the
constant polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
1 is monic, this semigroup is even a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
.


Partially ordered

The restriction of the divisibility relation to the set of all monic polynomials (over the given ring) is a
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
, and thus makes this set to a poset. 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 elements other than 1.


Polynomial equation solutions

In other respects, the properties of monic polynomials and of their corresponding monic polynomial equations depend crucially on the coefficient ring ''A''. If ''A'' is a field, then every non-zero polynomial ''p'' has exactly one associated 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 is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
coefficients cannot have rational 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 solutions. The roots of monic polynomials with integer coefficients are called algebraic integers. The solutions to monic polynomial equations over an integral domain are important in the theory of integral 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'': : 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 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 In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
'' over ''A''. If here A=\mathbb (the ring of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s) and B=\mathbb (the field of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s), then ''C'' is the ring of '' algebraic integers''.


Irreduciblity

If is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, the number of monic irreducible polynomials of degree over a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
\mathrm(p) 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 (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, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
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