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:$y^4+frac\{xy\}\{2\}=frac\{x^3\}\{3\}-xy^2+y^2-frac\{1\}\{7\}$
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:$42y^4+21xy=14x^3-42xy^2+42y^2-6$

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")).

ee also

* Algebraic function
* Algebraic number
* Algebraic geometry
* Galois theory