PlanetMath: binomial equation

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binomial equation

(Definition)

Binomial equation is an algebraic equation of the simple type

$\displaystyle x^n-a = 0$

where

belongs to a certain field (or sometimes to a certain ring) and

is the unknown (or the indeterminate) of the equation. Solving the binomial equation means taking the

The binomial equation is written also

$\displaystyle x^n = a.$

Such a binomial equation may be examined in a group, too.

A special case of the binomial equation is the cyclotomic equation

$\displaystyle x^n-1 = 0.$

This name comes from the fact that the roots of the equation divide the unit circle in the complex plane into

equally long arcs (Greek $\varkappa\acute{\upsilon}\varkappa\lambda{o}\varsigma$ `circle', $\tau\acute{o}\mu{o}\varsigma$ `part').

"binomial equation" is owned by pahio.

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Also defines:

cyclotomic equation

This object's parent.

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Cross-references: circle, arcs, complex plane, unit circle, group, equation, indeterminate, ring, field, integer, positive, algebraic equation
There are 3 references to this entry.

This is version 6 of binomial equation, born on 2008-01-29, modified 2008-01-31.
Object id is 10223, canonical name is BinomialEquation.
Accessed 661 times total.

Classification:

AMS MSC:	12E05 (Field theory and polynomials :: General field theory :: Polynomials )
	11C08 (Number theory :: Polynomials and matrices :: Polynomials)

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