- Algebraic equation
In

mathematics , an**algebraic equation**over a given field is anequation of the form:$P\; =\; Q$where "P" and "Q" are (possibly multivariate)polynomial s 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

coefficient s areinteger s (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 (becausesine ,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 offormal 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

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