In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an algebraic equation or polynomial equation is an
equation of the form
:
where ''P'' is 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 example ...
with
coefficients in some
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 grass ...
, often the field of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s. For many authors, the term ''algebraic equation'' refers only to ''univariate equations'', that is polynomial equations that involve only one
variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
. On the other hand, a polynomial equation may involve several variables. In the case of several variables (the ''multivariate'' case), the term ''polynomial equation'' is usually preferred to ''algebraic equation''.
For example,
:
is an algebraic 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 language ...
coefficients and
:
is a multivariate polynomial equation over the rationals.
Some but not all polynomial equations with
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
coefficients have a solution that is an
algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations ( addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). ...
that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be
solved algebraically). This can be done for all such equations of
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
one, two, three, or four; but for degree five or more it can only be done for some equations,
not all. A large amount of research has been devoted to compute efficiently accurate approximations of the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
solutions of a univariate algebraic equation (see
Root-finding algorithm
In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers ...
) and of the common solutions of several multivariate polynomial equations (see
System of polynomial equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some field .
A ''solution'' of a polynomial system is a set of values for the ...
).
Terminology
The term "algebraic equation" dates from the time when the main problem of
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 ...
was to solve
univariate
In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariate ...
polynomial equations. This problem was completely solved during the 19th century; see
Fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
,
Abel–Ruffini theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means th ...
and
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
.
Since then, the scope of algebra has been dramatically enlarged. In particular, it includes the study of equations that involve
th roots and, more generally,
algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations ( addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). ...
s. This makes the term ''algebraic equation'' ambiguous outside the context of the old problem. So the term ''polynomial equation'' is generally preferred when this ambiguity may occur, specially when considering multivariate equations.
History
The study of algebraic equations is probably as old as mathematics: the
Babylonian mathematicians, as early as 2000 BC could solve some kinds of
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not q ...
s (displayed on
Old Babylonian
Old Babylonian may refer to:
*the period of the First Babylonian dynasty (20th to 16th centuries BC)
*the historical stage of the Akkadian language
Akkadian (, Akkadian: )John Huehnergard & Christopher Woods, "Akkadian and Eblaite", ''The Camb ...
clay tablet
In the Ancient Near East, clay tablets ( Akkadian ) were used as a writing medium, especially for writing in cuneiform, throughout the Bronze Age and well into the Iron Age.
Cuneiform characters were imprinted on a wet clay tablet with a sty ...
s).
Univariate algebraic equations over the rationals (i.e., with
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
coefficients) have a very long history. Ancient mathematicians wanted the solutions in the form of
radical expression
In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'':
:r^n = x,
where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A roo ...
s, like
for the positive solution of
. The ancient Egyptians knew how to solve equations of degree 2 in this manner. The Indian mathematician Brahmagupta (597–668 AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD, but written in words instead of symbols. In the 9th century
Muhammad ibn Musa al-Khwarizmi
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persians, Persian polymath from Khwarazm, who produced vastly influential works in Mathematics ...
and other Islamic mathematicians derived the
quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, g ...
, the general solution of equations of degree 2, and recognized the importance of the
discriminant. During the Renaissance in 1545,
Gerolamo Cardano
Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
published the solution of
Scipione del Ferro
Scipione del Ferro (6 February 1465 – 5 November 1526) was an Italian mathematician who first discovered a method to solve the depressed cubic equation.
Life
Scipione del Ferro was born in Bologna, in northern Italy, to Floriano and Filip ...
and
Niccolò Fontana Tartaglia
Niccolò Fontana Tartaglia (; 1499/1500 – 13 December 1557) was an Italian mathematician, engineer (designing fortifications), a surveyor (of topography, seeking the best means of defense or offense) and a bookkeeper from the then Republi ...
to
equations of degree 3 and that of
Lodovico Ferrari
Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italian mathematician.
Biography
Born in Bologna, Lodovico's grandfather, Bartolomeo Ferrari, was forced out of Milan to Bologna. Lodovico settled in Bologna, and he began his ...
for
equations of degree 4. Finally
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
proved, in 1824, that
equations of degree 5 and higher do not have general solutions using radicals.
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
, named after
Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
, showed that some equations of at least degree 5 do not even have an idiosyncratic solution in radicals, and gave criteria for deciding if an equation is in fact 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 (that is, with
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
coefficients).
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
was introduced by
Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
to specify criteria for deciding if an algebraic equation may be solved in terms of radicals. In
field theory, an
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
is an extension such that every element is a root of an algebraic equation over the base field.
Transcendental number theory
Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways.
Transcendence
...
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
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
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
solutions
Solution may refer to:
* Solution (chemistry), a mixture where one substance is dissolved in another
* Solution (equation), in mathematics
** Numerical solution, in numerical analysis, approximate solutions within specified error bounds
* Soluti ...
. In particular the equation
is equivalent to
. 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
coefficients are
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 language ...
s. For example, multiplying through by 42 = 2·3·7 and grouping its terms in the first member, the previously mentioned polynomial equation
becomes
:
Because
sine,
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
, and 1/''T'' are not polynomial functions,
:
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
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and ...
s in the variable ''T''.
Theory
Polynomials
Given an equation in unknown
:
,
with coefficients in 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 grass ...
, one can equivalently say that the solutions of (E) in are the roots in of the polynomial
: