Algebraic equation
Encyclopedia
In mathematics
, an algebraic equation, also called polynomial equation over a given field
is an equation
of the form
where P and Q are (possibly multivariate) polynomial
s over that field. For example
is an algebraic equation over the rationals.
Two equations are equivalent if they have the same set of solutions
. In particular the equation
is equivalent with 
. It follows that the study of algebraic equations is equivalent to the study of polynomials.
An algebraic 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 algebraic equation above becomes the algebraic equation
Although the equation
is not an algebraic equation in four variables (x, y, z and T) over the rational numbers (because sine
, exponentiation
and 1/T are not polynomial functions). It is an algebraic equation in the three variables x, y, and z over Q((T)), the field of formal Laurent series in T over the rational numbers. Indeed, the coefficients
1/T and -2 are all elements of Q((T)).
As for any equation, the solutions of an equation are the values of the variables for which the equation is true, but for algebraic equations there 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 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 look for solutions in the field of complex numbers; the fundamental theorem of algebra
asserts that a non constant equation has always such solutions. Again, one may also be interested only in the real solutions.
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 expressions, like
for the positive solution of 
. The ancient Egyptians knew how to solve equations of degree 2 in this manner. During the Renaissance, Gerolamo Cardano
has found the solution of the equation of degree 3
and Lodovico Ferrari
has solved the equation of degree 4
. Finally Niels Henrik Abel
has proved in 1824 that the equation of degree 5
and the equations of higher degree are not always solvable using radicals. Galois theory
, named after Évariste Galois
, were introduced to give criteria deciding if an equation is solvable using radicals.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an algebraic equation, also called polynomial equation over a given field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
is an equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
of the form
where P and Q are (possibly multivariate) polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s over that field. For example
is an algebraic equation over the rationals.
Two equations are equivalent if they have the same set of solutions
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
. In particular the equation
is equivalent with . It follows that the study of algebraic equations is equivalent to the study of polynomials.An algebraic equation over the rationals can always be converted to an equivalent one in which the coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s are integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s. For example, multiplying through by 42 = 2·3·7 and grouping its terms in the first member, the algebraic equation above becomes the algebraic equation
Although the equation
is not an algebraic equation in four variables (x, y, z and T) over the rational numbers (because sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
, exponentiation
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
and 1/T are not polynomial functions). It is an algebraic equation in the three variables x, y, and z over Q((T)), the field of formal Laurent series in T over the rational numbers. Indeed, the coefficients
1/T and -2 are all elements of Q((T)).
As for any equation, the solutions of an equation are the values of the variables for which the equation is true, but for algebraic equations there 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 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
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
. One may also look for solutions in the field of complex numbers; the fundamental theorem of algebra
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
asserts that a non constant equation has always such solutions. Again, one may also be interested only in the real solutions.
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 expressions, like
for the positive solution of . The ancient Egyptians knew how to solve equations of degree 2 in this manner. During the Renaissance, Gerolamo CardanoGerolamo Cardano
Gerolamo Cardano was an Italian Renaissance mathematician, physician, astrologer and gambler...
has found the solution of the equation of degree 3
Cubic function
In mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...
and Lodovico Ferrari
Lodovico Ferrari
Lodovico Ferrari was an Italian mathematician.Born in Milan, Italy, grandfather, Bartholomew Ferrari was forced out of Milan to Bologna. He settled in Bologna, Italy and he began his career as the servant of Gerolamo Cardano. He was extremely bright, so Cardano started teaching him mathematics...
has solved the equation of degree 4
Quartic function
In mathematics, a quartic function, or equation of the fourth degree, is a function of the formf=ax^4+bx^3+cx^2+dx+e \,where a is nonzero; or in other words, a polynomial of degree four...
. Finally Niels Henrik Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
has proved in 1824 that the equation of degree 5
Quintic equation
In mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...
and the equations of higher degree are not always solvable using radicals. Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
, named after Évariste Galois
Évariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
, were introduced to give criteria deciding if an equation is solvable using radicals.
See also
- Algebraic functionAlgebraic functionIn mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
- Algebraic numberAlgebraic numberIn mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
- Algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
- Galois theoryGalois theoryIn mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
- Root finding
- System of polynomial equations