Algebraic equation

In mathematics, an algebraic equation, also called polynomial equation over a given field is an equation of the form

$P = Q$

where P and Q are (possibly multivariate) polynomials over that field. For example

$y^4+\frac{xy}{2}=\frac{x^3}{3}-xy^2+y^2-\frac{1}{7}$

is an algebraic equation over the rationals.

Two equations are equivalent if they have the same set of solutions. In particular the equation $P = Q$ is equivalent with $P-Q = 0$. 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 coefficients are integers. 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

$42y^4+21xy-14x^3+42xy^2-42y^2+6=0$

Although the equation

$e^T x^2+\frac{1}{T}xy+\sin(T)z -2 =0$

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

$e^T=1+T+\frac{T^2}{2!}+\frac{T^3}{3!}+\cdots$

$\sin(T)=T - \frac{T^3}{3!} + \frac{T^5}{5!} - \frac{T^7}{7!} + \cdots$

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 $x=\frac{1+\sqrt{5}}{2}$ for the positive solution of $x^2+x-1=0$. 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.

References

• Weisstein, Eric W., "Algebraic Equation" from MathWorld.

See also

• Algebraic function
• Algebraic number
• Algebraic geometry
• Galois theory
• Root finding
• System of polynomial equations