Algebraic equation

In mathematics, an algebraic polynomial over a given field is a polynomial with coefficients in that field. In the simplest case, which is often meant when not otherwise specified, the field is Q, the field of rational numbers, and the algebraic polynomials are those with rational coefficients. For example

$-7x^{3}+{\begin{matrix}{\frac {2}{3}}\end{matrix}}x^{2}-5x+3$

is an algebraic polynomial over the rationals.

An algebraic equation over a given field is an equation of the form

$P=0$

where $P$ is a (possibly multivariate) algebraic polynomial over that field. For example

$x^{2}+3xy-4y^{2}+1=0$

is an algebraic equation over the rationals.

Note that an algebraic equation over the rationals can always be converted to one in which the coefficients are integers. For example, multiplying through by three, the first algebraic polynomial above forms the algebraic equation

$-21x^{3}+2x^{2}-15x+9=0.$

The standard form of such an equation, however, has leading coefficient unity. If all the other coefficients of this form are integers, the roots of the equation are algebraic integers.

Although the equation

$e^{T}x^{2}+{\frac {1}{T}}xy-5\sin(T)z-2=0$

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 over $\mathbb {Q} ((T))$, the field of formal Laurent series in $T$ over the rational numbers.

See also

• Algebraic number
• Algebraic Geometry
• Galois Theory