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

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

An algebraic equation is an equation of the form

$P\; =\; 0$

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

$x^2\; +\; 3xy\; -\; 4y^2\; +\; 1\; =\; 0$

Note that an algebraic equation 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.

See also

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• Algebraic number
• Algebraic Geometry
• Galois Theory

Polynomials | Equations

Algebraische Gleichung | Equazione algebrica | Równanie algebraiczne | Phương trình đại số | Алгебраїчне рівняння