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# binomial equation

Binomial equation is an algebraic equation of the simple type

$x^{n}-a=0$ |

where $n$ is a positive integer, $a$ belongs to a certain field (or sometimes to a certain ring) and $x$ is the unknown (or the indeterminate) of the equation. Solving the binomial equation means taking the $n$th root of $a$.

The binomial equation is written also

$x^{n}=a.$ |

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

A special case of the binomial equation is the cyclotomic equation

$x^{n}-1=0.$ |

This name comes from the fact that the roots of the equation divide the unit circle in the complex plane into $n$ equally long arcs (Greek $\varkappa\acute{\upsilon}\varkappa\lambda{o}\varsigma$ ‘circle’, $\tau\acute{o}\mu{o}\varsigma$ ‘part’).

Defines:

cyclotomic equation

Related:

Binomial, CalculatingTheNthRootsOfAComplexNumber, RootOfUnity

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

11C08*no label found*12E05

*no label found*

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## Comments

## Binomial equation name

Jussi,

Do you have a reference for this term? I've never seen it used (at least not for this).

Roger

## Re: Binomial equation name

Dear Roger,

I know that the term "binomial equation" is rare in the English language, at least in America, but since it is quite good term, I wanted to tell it in PM. Unfortenately, I have no algebra books in English. Here some Engl. references in internet:

http://thesaurus.maths.org/mmkb/entry.html?action=entryById&id=3536

http://www.answers.com/topic/binomial-equation?cat=technology

http://en.wikipedia.org/wiki/Joseph_Louis_Lagrange

In Europe, the equivalents of the term are more used, e.g. in German "binomische Gleichung", ref.

http://www-user.tu-chemnitz.de/~syha/lehre/baI/baI.pdf

in Swedish "binomisk ekvation", ref.

www.matematik.lu.se/matematiklth/personal/sigrid/analys1/lektion19.ps

Jussi

## Re: Binomial equation name

Harry Hochstadt in his book on special functions uses the same sort

of terminology when referring to Mellin's work on integral

representations and differential equations for certain algebraic

functions such as the general solution of the quintic. (Again

the Finnish connection!) To be sure, there the term used is

"trinomial equation", but that is because the equation considered

there has three terms as opposed to two.

While this might sound unusual to Americans --- here, the term

"binomial" is usually appears in the context of binomial

expansions and binomial coefficients, the usage is quite

logical. Just as the words "monomial" and "polynomial" refer

to algebraic expressions with one or many terms respectively,

so too the terms "binomial", "trinomial" are used to refer to

expressions with two and three terms, respectively and the

terms "binomial equation" and "trinomial equation" refer to

equations gotten by setting such expressions to zero.

By the way, Jussi, could you add the form of Mellin's inversion

formula for his transform to your entry on that topic? I would

like to attach an entry explaining Mellin's ingenious use of

inverting a linear transform to the solve non-linear problem

of inverting a trinomial or other algebraic, even transcendental,

function.

## Re: Binomial equation name

> By the way, Jussi, could you add the form of Mellin's

> inversion formula for his transform to your entry on

> that topic? I would like to attach an entry explaining

> Mellin's ingenious use of inverting a linear transform

> to the solve non-linear problem of inverting a trinomial

> or other algebraic, even transcendental, function.

Raymond,

There is no entry on Mellin's transform and its inversion in PM, but a request on it. I cannot write such an entry since I don't know this subject, but I understand that you know. I would be very happy if you could write on those things. (In the biography of Mellin I have mentioned the transform and its inverse.)

Jussi

## Re: Binomial equation name

One Mellin transform entry coming right up!

## Re: Binomial equation name

> One Mellin transform entry coming right up!

Raymond,

I see you have been very hectic and have not yet had time for writing on the transform. I hope you have not forgot the thing =o)

Jussi

## Re: Binomial equation name

No, I did not forget. As you said, I am busy.

This is one of many items on my list of things

to do, including also, for instance, writing

more about C*-algebras and finishing the treatment

of Riemann surfaces from the standpoint of sheaf theory.

## Re: Binomial equation name

Oh, you have so much works! I have only one (a linguistic one).