Algebraic equation

In mathematics, the term algebraic equation in a narrower and a wider sense used.

Closer importance

In a narrower sense is meant by an algebraic equation of degree over a ring or body an equation of the form

With a polynomial of degree n over, so an equation of the form

With coefficients and

If not specified in more detail, so usually the real numbers are meant, so for example, the equation

In the case of rational numbers, the equation by multiplying by the least common multiple of the denominators of the coefficients can always be converted into an equivalent equation with integer coefficients, we obtain approximately

For the above example.

Each solution of an algebraic equation over the rational numbers is called algebraic number; with algebraic equations over an arbitrary field, the solutions are called algebraic elements. This term expresses that such a solution does not need to lie in the ring or the body, from which the coefficients of the equations, but rather only in a suitable extension ring or body.

Every algebraic equation of positive degree with real or complex coefficients has at least one complex solution. This is the statement of the Fundamental Theorem of Algebra.

The solutions of an algebraic equation with real coefficients are real or pairwise complex conjugate.

One can also define algebraic equations for functions. Taking as a coefficient ring the ring

Of continuous functions over the positive half- axis and is denoted by x is given by x (t ) = t for all t defined identical function, the square root function is a solution of the algebraic equation

Such an approach is necessary in order to investigate solutions of certain algebraic equations.

Further meaning

In a broader sense, algebraic equation is also used as differentiation from differential equations. We are talking, for example, in the differential-algebraic equation

( Are given functions of a subset of by; are unknown functions of a subset of after ) the second equation as an algebraic equation ( regardless of whether algebraically in the narrower sense), to distinguish it from the first equation, the differential equation.

• Algebra