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In [[algebra]], a '''monic polynomial''' is a [[polynomial]]


:<math>c_nx^n+c_{n-1}x^{n-1}+\cdots+c_2x^2+c_1x+c_0</math>
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in which the leading coefficient ''c''<sub>''n''</sub> is equal to&nbsp;1.
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== Univariate polynomials ==
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If a [[polynomial]] has only one [[indeterminate (variable)|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
 
 
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: ''c''<sub>''n''</sub> ≠ 0, ''c''<sub>''n''&minus;1</sub>, ..., ''c''<sub>2</sub>, ''c''<sub>1</sub> and ''c''<sub>0</sub>
 
 
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Here the term ''c''<sub>''n''</sub>''x''<sup>''n''</sup> is called the ''leading term'', and its coefficient ''c''<sub>''n''</sub> the ''leading coefficient''; if the leading coefficient {{nowrap|is 1}}, the univariate polynomial is called '''monic'''.
 
 
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===Examples===
 
* [[Complex quadratic polynomial]]s
</ul>
 
=== Properties ===
 
====Multiplicatively closed====
The set of all monic polynomials (over a given (unitary) [[ring (mathematics)|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]] 1 is monic, this semigroup is even a [[monoid]].
 
====Partially ordered====
The restriction of the [[divisibility (ring theory)|divisibility]] relation to the set of all monic monomials (in the given ring) is a [[partial order]], 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 monomials ''p'' and ''q'', then ''p'' and ''q'' must be equal. The corresponding property is not true for polynomials in general, if the ring contains other [[invertible element]]s than 1.
 
====Polynomial equation solutions====
In other respects, the properties of monic polynomials and of their corresponding monic [[polynomial equation]]s depend crucially on the coefficient ring ''A''. If ''A'' is a [[field (algebra)|field]], then every non-zero polynomial ''p'' has exactly one [[associated element|associated]] monic polynomial ''q''; actually, ''q'' is ''p'' divided with its leading coefficient. In this manner, then, any non-trivial polynomial equation ''p''(''x'')&nbsp;=&nbsp;0 may be replaced by an equivalent monic equation ''q''(''x'')&nbsp;=&nbsp;0. E.g., the general real second degree equation—
:<math>\ ax^2+bx+c = 0</math> (where <math> a \neq 0</math>)
may be replaced by
:<math>\ x^2+px+q = 0</math>,
by putting &nbsp;''p''&nbsp;=&nbsp;''b''/''a''&nbsp; and &nbsp;''q''&nbsp;=&nbsp;''c''/''a''. Thus, 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>
 
The general quadratic solution formula is then the slightly more simplified form of:
:<math>x = \frac{1}{2} \left( -p \pm \sqrt{p^2 - 4q} \right).</math>
 
=====Integrality=====
On the other hand, if the coefficient ring is not a field, there are more essential differences.  E.g., a monic polynomial equation with [[integer]] coefficients cannot have other [[Rational number|rational]] solutions than integer solutions. Thus, the equation
:<math>\ 2x^2+3x+1 = 0</math>
possibly might have some rational root, which is not an integer, (and incidentally it does have ''inter alia'' the root &minus;1/2); while the equations
:<math>\ x^2+5x+6 = 0</math>
and
:<math>\ x^2+7x+8 = 0</math>
only may have integer solutions or [[irrational number|irrational]] solutions.
 
The roots of monic polynomial with integer coefficients are called [[algebraic integer]]s.
 
The solutions to monic polynomial equations over an [[integral domain]] are important in the theory of [[integral extension]]s and [[integrally closed domain]]s, 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'':
:<math> C := \{b \in B : \exists\, p(x) \in A[x]\,, \hbox{ which is monic and such that } p(b) = 0\}\,.</math>
The set ''C'' contains ''A'', since any ''a''&nbsp;∈&nbsp;''A'' satisfies the equation ''x''&nbsp;−&nbsp;''a''&nbsp;=&nbsp;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 ''[[integrality|integral]]'' over ''A''. If here <math>A=\mathbb{Z}</math> (the ring of [[integer]]s) and <math>B=\mathbb{C}</math> (the field of [[complex number]]s), then ''C'' is the ring of ''[[algebraic integers]]''.
 
== 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
:<math>\ p(x,y) = 2xy^2+x^2-y^2+3x+5y-8</math>
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'':
:<math>p(x,y) = 1\cdot x^2 + (2y^2+3) \cdot x + (-y^2+5y-8)</math>;
but ''p''(''x'',''y'') is not monic as an element in  '''R'''[''x''][''y''], since then the highest degree coefficient (i.e., the ''y''<sup>2</sup> coefficient) is &nbsp;2''x''&nbsp;&minus;&nbsp;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''<sub>1</sub>'',...,x<sub>n</sub>'') 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]] generated by ''x''<sub>1</sub>'',...,x<sub>n</sub>'', 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.
 
== References ==
* {{cite book |last=Pinter |first=Charles C. |title=A Book of Abstract Algebra |year=2010 |origyear=Unabridged republication of the 1990 second edition of the work originally published in 1982 by the McGraw–Hill Publishing Company |publisher=Dover |isbn=978-0486474175}}
 
[[Category:Polynomials]]

Latest revision as of 02:35, 27 December 2014

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