Algebraic equation

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In mathematics , the term algebraic equation is used in a narrower and a broader meaning.

Narrower meaning

In a narrower sense, an algebraic equation of degree over a ring or body is an equation of form ${\displaystyle n}$ ${\displaystyle K}$

${\displaystyle P_{n}(x)=0}$

with a polynomial -th degree over , i.e. an equation of shape ${\displaystyle P_{n}(x)}$ ${\displaystyle n}$ ${\displaystyle K}$

${\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{1}x+a_{0}=\sum _{i=0}^{n}a_{i}x^{i}=0}$

with coefficients from and ${\displaystyle a_{i}}$ ${\displaystyle K}$ ${\displaystyle a_{n}\neq 0.}$

If it is not specified more precisely, the real numbers are usually meant, for example the equation ${\displaystyle K}$

${\displaystyle -7x^{3}+{\tfrac {2}{3}}x^{2}-5x+3=0.}$

In the case of rational numbers, the equation can always be converted into an equivalent equation with integer coefficients by multiplying it by the lowest common multiple of the denominators , for example

${\displaystyle -21x^{3}+2x^{2}-15x+9=0}$

for the above example.

Every solution of an algebraic equation over the rational numbers is called an algebraic number ; for algebraic equations over any field, the solutions are called algebraic elements . This designation expresses that such a solution does not have to lie in the ring or body from which the coefficients of the equation originate, but 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. That is what the fundamental theorem of algebra says .

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

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

${\displaystyle R=C(\mathbb {R} _{+},\mathbb {R} )}$

of the continuous functions over the positive semi-axis and denotes with x the identical function defined by x (t) = t for all t , then the square root function is a solution of the algebraic equation

${\displaystyle y(t)^{2}-x(t)=0.}$

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

Further meaning

In a broader sense, algebraic equation is also used as a demarcation from differential equations . This is the name given to the algebro differential equation , for example

${\displaystyle {\begin{matrix}{\dot {x}}_{1}(t)&=&f_{1}(x_{1}(t),x_{2}(t),t)\\0&=&f_{2}(x_{1}(t),x_{2}(t),t)\end{matrix}}}$

( are given functions of a subset of after ; are functions of a subset of after ) the second equation as an algebraic equation (regardless of whether it is algebraic in the narrower sense) in order to distinguish it from the first equation, the differential equation . ${\displaystyle f_{1},f_{2}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle x_{1},x_{2}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle f_{2}}$