# Algebraic equation

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:$y^4+frac\left\{xy\right\}\left\{2\right\}=frac\left\{x^3\right\}\left\{3\right\}-xy^2+y^2-frac\left\{1\right\}\left\{7\right\}$
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:$e^T x^2+frac\left\{1\right\}\left\{T\right\}xy+sin\left(T\right)z -2 =0$
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

:$e^T=1+T+frac\left\{T^2\right\}\left\{2!\right\}+frac\left\{T^3\right\}\left\{3!\right\}+cdots$:$sin\left(T\right)=T - frac\left\{T^3\right\}\left\{3!\right\} + frac\left\{T^5\right\}\left\{5!\right\} - frac\left\{T^7\right\}\left\{7!\right\} + cdots$

1/"T" and -2 are all elements of Q(("T")).

ee also

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

Wikimedia Foundation. 2010.

Поможем решить контрольную работу

### Look at other dictionaries:

• algebraic equation — Math. an equation in the form of a polynomial having a finite number of terms and equated to zero, as 2x3 + 4x2 x + 7 = 0. * * * Mathematical statement of equality between algebraic expressions. An expression is algebraic if it involves a finite… …   Universalium

• algebraic equation — noun A mathematical equation in which one or both sides is an algebraic expression, such as 2x + 7y = 3 …   Wiktionary

• algebraic equation — noun : an equation obtained by equating to zero a sum of a finite number of terms each one of which is a product of positive integral powers (including the zero power) of the variables * * * Math. an equation in the form of a polynomial having a… …   Useful english dictionary

• Differential algebraic equation — In mathematics, differential algebraic equations (DAEs) are a general form of (systems of) differential equations for vector–valued functions x in one independent variable t, where is a vector of dependent variables and the system has as many… …   Wikipedia

• Equation — E*qua tion, n. [L. aequatio an equalizing: cf. F. [ e]quation equation. See {Equate}.] 1. A making equal; equal division; equality; equilibrium. [1913 Webster] Again the golden day resumed its right, And ruled in just equation with the night.… …   The Collaborative International Dictionary of English

• Equation box — Equation E*qua tion, n. [L. aequatio an equalizing: cf. F. [ e]quation equation. See {Equate}.] 1. A making equal; equal division; equality; equilibrium. [1913 Webster] Again the golden day resumed its right, And ruled in just equation with the… …   The Collaborative International Dictionary of English

• Equation clock — Equation E*qua tion, n. [L. aequatio an equalizing: cf. F. [ e]quation equation. See {Equate}.] 1. A making equal; equal division; equality; equilibrium. [1913 Webster] Again the golden day resumed its right, And ruled in just equation with the… …   The Collaborative International Dictionary of English

• Equation of a curve — Equation E*qua tion, n. [L. aequatio an equalizing: cf. F. [ e]quation equation. See {Equate}.] 1. A making equal; equal division; equality; equilibrium. [1913 Webster] Again the golden day resumed its right, And ruled in just equation with the… …   The Collaborative International Dictionary of English

• Equation of equinoxes — Equation E*qua tion, n. [L. aequatio an equalizing: cf. F. [ e]quation equation. See {Equate}.] 1. A making equal; equal division; equality; equilibrium. [1913 Webster] Again the golden day resumed its right, And ruled in just equation with the… …   The Collaborative International Dictionary of English

• Equation of payments — Equation E*qua tion, n. [L. aequatio an equalizing: cf. F. [ e]quation equation. See {Equate}.] 1. A making equal; equal division; equality; equilibrium. [1913 Webster] Again the golden day resumed its right, And ruled in just equation with the… …   The Collaborative International Dictionary of English