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To reach it in excel, copy-paste this continued plan into corpuscle B1. If you do again access an almost all time in abnormal all over corpuscle A1, the discount in treasures will come to pass in B1.<br><br>
In [[mathematics]], an '''algebraic equation''' or '''polynomial equation''' is an [[equation]] of the form
:<math>P = Q</math>
where ''P'' and ''Q'' are [[polynomial]]s with coefficients in some [[field (mathematics)|field]], often the field of the [[rational number]]s. For most authors, an algebraic equation is ''univariate'', which means that it involves only one [[variable (mathematics)|variable]]. On the other hand, a polynomial equation may involve several variables, in which case it is called ''multivariate'' and the term ''polynomial equation'' is usually preferred to ''algebraic equation''.


For example,
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:<math>x^5-3x+1=0</math>
is an algebraic equation with integer coefficients and
:<math>y^4+\frac{xy}{2}=\frac{x^3}{3}-xy^2+y^2-\frac{1}{7}</math>
is a multivariate polynomial equation over the rationals.
 
Some but not all polynomial equations with [[Rational number|rational coefficients]] have a solution that is an [[algebraic expression]] with a finite number of operations involving just those coefficients (that is, can be [[Algebraic solution|solved algebraically]]). This can be done for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some equations but [[Abel-Ruffini theorem|not for all]]. A large amount of research has been devoted to compute efficiently accurate approximations of the [[real number|real]] or [[complex number|complex]] solutions of an univariate algebraic equation (see [[Root-finding algorithm]]) and of the common solutions of several multivariate polynomial equations (see [[System of polynomial equations]]).
 
== History ==
 
The study of algebraic equations is probably as old as mathematics: the [[Babylonian mathematics|Babylonian mathematicians]], as early as 2000 BC could solve some kind of [[quadratic equation]]s (displayed on [[First Babylonian Dynasty|Old Babylonian]] [[clay tablet]]s).
 
The algebraic equations over the rationals with only one variable are also called univariate equations. They have a very long history. Ancient mathematicians wanted the solutions in the form of [[radical expression]]s, like <math>x=\frac{1+\sqrt{5}}{2}</math> for the positive solution of <math>x^2-x-1=0</math>. The ancient Egyptians knew how to solve equations of degree 2 in this manner.  In the 9th century [[Muhammad ibn Musa al-Khwarizmi]] and other Islamic mathematicians derived the general [[quadratic formula]] and recognized the importance of the [[discriminant]].  During the Renaissance, [[Gerolamo Cardano]] found the solution to [[Cubic function|equations of degree 3]] and [[Lodovico Ferrari]] solved [[Quartic function|equations of degree 4]]. Finally [[Niels Henrik Abel]] proved, in 1824, that [[quintic equation|equations of degree 5]] and equations of higher degree are not always solvable using radicals. [[Galois theory]], named after [[Évariste Galois]], was introduced to give criteria deciding if an equation is solvable using radicals.
 
== Areas of study ==
 
The algebraic equations are the basis of a number of areas of modern mathematics: [[Algebraic number theory]] is the study of (univariate) algebraic equations over the rationals. [[Galois theory]] has been introduced by [[Évariste Galois]] for getting criteria deciding if an algebraic equation may be solved in terms of radicals. In [[field theory (mathematics)|field theory]], an [[algebraic extension]] is an extension such that every element is a root of an algebraic equation over the base field. [[Transcendence theory]] is the study of the real numbers which are not solutions to an algebraic equation over the rationals. A [[Diophantine equation]] is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions. [[Algebraic geometry]] is the study of the solutions in an [[algebraically closed field]] of multivariate polynomial equations.
 
Two equations are equivalent if they have the same set of [[Equation|solutions]]. In particular the equation <math>P = Q</math> is equivalent with <math>P-Q = 0</math>. It follows that the study of algebraic equations is equivalent to the study of polynomials.
 
A polynomial equation over the rationals can always be converted to an equivalent one in which the [[coefficient]]s are [[integer]]s. For example, multiplying through by 42 = 2·3·7 and grouping its terms in the first member, the polynomial equation above becomes
:<math>42y^4+21xy-14x^3+42xy^2-42y^2+6=0.</math>
 
Because [[sine]], [[exponentiation]], and 1/''T'' are not polynomial functions,
:<math>e^T x^2+\frac{1}{T}xy+\sin(T)z -2 =0</math>
is ''not'' a polynomial equation in the four variables ''x'', ''y'', ''z'', and ''T'' over the rational numbers. However, it is a polynomial equation in the three variables ''x'', ''y'', and ''z'' over the field of the [[elementary function]]s in the variable ''T''.
 
As for any equation, the solutions of an equation are the values of the variables for which the equation is true. For univariate algebraic equations these are also called '''roots''', even if, properly speaking, one should say ''the '''solutions''' of the algebraic equation'' ''P''=0 ''are the '''roots''' of the polynomial'' ''P''. When solving an equation, it is important to specify in which [[Set (mathematics)|set]] the solutions are allowed. For example, for an equation over the rationals one may look for solutions in which all the variables are integers. In this case the equation is a [[diophantine equation]]. One may also be interested only in the real solutions. However, for univariate algebraic equations, the number of solutions is finite and all solutions, are contained in any [[algebraically closed field]] containing the coefficients, for example, the field of complex numbers in case of equations over the rationals. It follows that without precision "root" and "solution" usually mean "solution in an algebraically closed field".
 
==See also==
* [[Algebraic function]]
* [[Algebraic number]]
* [[Root finding]]
 
==References==
* {{springer|title=Algebraic equation|id=p/a011480}}
* {{MathWorld|title=Algebraic Equation|urlname=AlgebraicEquation}}
 
{{DEFAULTSORT:Algebraic Equation}}
[[Category:Polynomials]]
[[Category:Equations]]

Latest revision as of 04:59, 6 November 2014

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