Algebraic equation

In mathematics, an algebraic 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

is an algebraic equation over the rationals.

Note that an algebraic equation over the rationals can always be converted to an equivalent one in which the coefficients are integers (where equivalence refers to the fact that the two equations will have the same solutions). For example, multiplying through by 42 = 2·3·7, the algebraic equation above becomes the algebraic equation

42y⁴ + 21xy = 14x³ − 42xy² + 42y² − 6

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)).

See also

• Algebraic function
• Algebraic number
• Algebraic geometry
• Galois theory