Algebraic equation

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If, in mathematics, I express it with the generic name of the equation referred to in the form that linked the multinomial expression by an equal sign (is multivariate generally) the algebraic equation (だいすうほうていしき British: algebraic equation) in a ceremony

${\displaystyle \sum a_{e_{1}e_{2}\ldots e_{m}}x_{1}^{e_{1}}x_{2}^{e_{2}}\cdots x_{m}^{e_{m}}=0}$

It is a thing expressed by の form. In other words, the algebraic equation is a mathematical object describing the point zero of the multinomial expression.

Table of contents

Summary

The algebraic equation is important in mathematics as the geometric problem for the area and arithmetic problems such as the Diophantine equation from ancient times; was studied. An algebraic equation and examples of the study include an + bn = cⁿ which Fermat considered as a problem for group (a, b, c) (Pythagoras number) of the natural number to satisfy Pythagorean theorem a2 + b2 = c² and the generalization in the 17th century. About the latter example, a claim not to exist except a self-evident thing (in the case of abc = 0) is known to the group of the natural number to satisfy this as the last theorem of Fermat.

In addition, geometric consideration has been accomplished including the classification theory of a quadric curve and the conicoid by Euler and others since Descartes invented the orthogonality coordinate system about the multivariate algebraic equation.

After the 19th century, as for the study on root of 1 variable multinomial expression, it was a portent of the abstract algebra including the invention of the theory of groups by Galois, and algebra geometry is established as a field studying the point zero of the multivariate multinomial expression (or pro-an equation) geometrically in the first half in the 20th century. The last theorem of above-mentioned Fermat was solved after time more than 300 years by the presentation of the problem, but therefore knowledge of the high mathematics including the algebra geometry was used.

I decide to leave a multivariate case to the item of the algebra geometry and explain the cause of the bodies such as Yuri such as several in detail about a coefficient and the algebraic equation of 1 variable to do in this clause mainly as follows. If, with the algebraic equation of 1 variable, I transpose it and arrange it

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

It is an equation expressed by form of () fixed number that each a_i is unrelated to (variable x). I assume it the degree of this algebraic equation with the degree of the multinomial expression of the left side of a go board then. In other words, it is said that it is a time equation of n-th degree of a_n ≠ 0.

• Linear equation ax + b = 0 (a ≠ 0)
• Quadratic equation ax2 + bx + c = 0 (a ≠ 0)
• Cubic equation ax3 + bx2 + cx + d = 0 (a ≠ 0)
• Biquadratic ax4 + bx3 + cx2 + dx + e = 0 (a ≠ 0)
• The algebraic equations more than the fifth "are not known to be solved algebraically". (Theorem of Abel - ルフィニ)

Concepts

Root

I assume f(x) a multinomial expression about x. A solution of algebraic equation f(x) = 0 of the root (root which does not come) in particular (it is said with both "the root of the equation and" "the root of the multinomial expression").

It is the equivalent that x = α being a root of multinomial expression f(x) and multinomial expression f(x) have x-α to a factor by factor theorem. In integer k and multinomial expression g(x) regular for multinomial expression f(x)

${\displaystyle f(x)=(x-\alpha) ^{k}g(x),\quad g(\alpha) \neq 0}$ 

When a thing satisfying を exists, I say α with a point zero of k equal root (じゅうこん, multiple root) or the k f(x) digit and say k with an overlap degree (ちょうふくど, multiplicity) of root α or an order of magnitude. But I say a single root (simple root, simple zero where a sputum does not come to) at the time of k = 1. In addition, there is a case to point to only with a doubleness root when I name the cause that is not a single root generally by context when I call it merely equal root. If put it to an overlap degree; with the root of the algebraic equation

${\displaystyle a(x-\alpha _{1})(x-\alpha _{2})\cdots (x-\alpha _{n})=0}$ 

となるときの α ₁, α ₂, ...When it is ,α_n, I am expressed in other words.

A root of Clause 2 multinomial expression xn-a is called 冪根 in particular.

Algebraic figure

The algebraic equation does not come loose generally in K when I assume a multinomial expression coefficient body of the left side of a go board K, but can show existence of extended body L of K including the root when I was given one algebraic equation. Furthermore, an algebra shut parcel of K exists except a difference of the same type uniquely. I fix 代数閉包 K ∧ one. That when former x of K ^∧ becomes the root of the algebraic equation of a certain K coefficient, x is algebraic in K (I leave it to a body theory in detail). Particularly, it is said that z is an algebraic figure if complex number z is algebraic in Yuri several Q.

整方程式

As for the root of the multinomial expression that is モニック which assumes ring R the ring of the coefficient (the most highly advanced coefficient 1), it is said that it is 整 on R. It is the equivalent to be 整 if R is a body and to be algebraic.

Elucidation of the algebraic equation

Summary

Methods to identify the root of the algebraic equation logically include a thing by "numerical elucidation" (approximate algorithm), a thing by "algebraic elucidation" (combination of addition, subtraction, multiplication and division operation and limited time of the operation to add 冪根 to), a thing by "elucidation (oval modular function, substitution to a super geometric progression, combination of limited times of the addition, subtraction, multiplication and division operation) of transcending it". Two latter is methods to show a thing called "the formula of the solution". In addition, call for all numerical analysis of them, and the numerical elucidation is used, for example, for the general equation including the equation including an exponential function and the logarithmic function as well as an algebraic equation widely.

Equations less than the fourth are known to have the formula of the solution by the algebraic elucidation. There is the formula of the solution by the method of transcending it in the equation that is more highly advanced than the fifth. It is that I am misunderstood well, but it points to the formula of the solution by the algebraic elucidation not existing that "generally do not come loose as for the fifth equation" that I is said generally and is that I only repeat operation to take addition, subtraction, multiplication and division operation and 冪乗根 from the coefficient of the algebraic equation that all algebraic figures think about in a limitedness time and am not provided. This is a fact shown by ルフィニ and Abel. The set of the whole algebraic figure is wide in this sense. I am apt to be confused by the name called the algebraic figure, but the algebraic figure is not necessarily only a thing provided by an algebraic method.

However, Galois foretells that general elucidation exists by the method of transcending it using the oval modular function and leaves it to the will. After death of Galois, the L meat led the formula of the solution of the fifth equation by the oval modular function.

In addition, it is thought because Abel studied the modular equation so that he had the official idea of the solution. The formula of the solution of the equation that is more highly advanced than the fifth is suggested variously to date by an L meat.

Because there is not much computational utility in the elucidation to depend formally of these solutions from the standpoint of the engineering, as for the equation that is more highly advanced than the third, elucidation by the numerical computation is common. I result in the issue of eigenvalue, and the algorithm of the eigenvalue calculation of the line may be used among them.

Official of the solution

I show the official summary of the solution as follows. See each article about the detailed contents.

Linear equation 

Linear equations are always solved in K without depending on coefficient body K.

Linear equation ${\displaystyle ax+b=0}$  ( ${\displaystyle a,b}$  は real number, ${\displaystyle a\neq 0}$  Solution of) ${\displaystyle x}$  Oh,

${\displaystyle x=-{\frac {b}{a}}}$  I can express と.

Quadratic equation

Quadratic equation ax2 + bx + c = 0 in the health that 標数 is not 2 comes loose in body F(a, b, c, √ D) which added the equilateral square root of coefficient a, b, c and discriminant D = b²-4ac in basic body F; and; at bottom (is known to be given in -b ±√ D)/2a.

Quadratic equation ${\displaystyle ax^{2}+bx+c=0}$  ( ${\displaystyle a,b,c}$  は real number, ${\displaystyle a\neq 0}$  Solution of) ${\displaystyle x}$  Oh,

${\displaystyle x={\frac {-b\pm {\sqrt {D}}}{2a}}}$  I can express と. But, ${\displaystyle D=b^{2}-4ac}$ 

Cubic equation

I can untie 1 empty cubic root, D in Q(³ √ ξ ₁, ³ √ ξ ₂, ω) in ω so that the algebraic elucidation of cubic equation ax3 + bx2 + cx + d = 0 is known as a formula of カルダノ if I choose suitable former ξ ₁, ξ ₂ as a thing of the discriminant of the cubic equation from Q(a, b, c, d,ω, √ D).

Cubic equation ${\displaystyle ax^{3}+bx^{2}+cx+d=0}$  ( ${\displaystyle a,b,c,d}$  は real number, ${\displaystyle a\neq 0}$  Solution of) ${\displaystyle x}$  Oh,

${\displaystyle x={\begin{cases}{\sqrt[{3}]{-{q \over 2}+{\sqrt {\left({q \over 2}\right)^{2}-\left({p \over 3}\right)^{3}}}}}+{\sqrt[{3}]{-{q \over 2}-{\sqrt {\left({q \over 2}\right)^{2}-\left({p \over 3}\right)^{3}}}}}-{b \over 3a}\\\omega {\sqrt[{3}]{-{q \over 2}+{\sqrt {\left({q \over 2}\right)^{2}-\left({p \over 3}\right)^{3}}}}}+\omega ^{2}{\sqrt[{3}]{-{q \over 2}-{\sqrt {\left({q \over 2}\right)^{2}-\left({p \over 3}\right)^{3}}}}}-{b \over 3a}\\\omega ^{2}{\sqrt[{3}]{-{q \over 2}+{\sqrt {\left({q \over 2}\right)^{2}-\left({p \over 3}\right)^{3}}}}}+\omega {\sqrt[{3}]{-{q \over 2}-{\sqrt {\left({q \over 2}\right)^{2}-\left({p \over 3}\right)^{3}}}}}-{b \over 3a}\end{cases}}}$  I can express と. But, ${\displaystyle {\begin{cases}p=-{1 \over 3}\left({b \over a}\right)^{2}+{c \over a}\\q=2\left({b \over 3a}\right)^{3}-{bc \over 3a^{2}}+{d \over a}\\\omega ={1+{\sqrt {3}}i \over 2}\end{cases}}}$ 

Biquadratic

The algebraic elucidation of biquadratic ax4 + bx3 + cx2 + dx + e = 0 is known as elucidation of the feh lari. This elucidation uses the ceremony of perfect square, and I change into form of ² = (linear equation) ² specifically (quadratic equation) and will untie it, but operation to untie a cubic equation in a process of this transformation is necessary.

The fifth equation

Because the formula of the solution using the oval modular function is complicated, I leave it in an outline. The fifth equation is transformed with x5-x-A = 0 by チルンハウス conversion (English version) (general form of the fifth equation). On the other hand, 4 roots of modulars provided by the fifth conversion of the oval function become the sixth equation called the modular equation. This equation is transformed in form of y5 + y-B = 0 by チルンハウス conversion (B becomes the algebraic expression of 4 roots of the number of the kinds of the oval function). In other words, the comparison between general form and between coefficients modular equation of the fifth equation becomes the biquadratic. On the other hand, as for the solution of the modular equation, the formula of the fifth equation is provided by an oval modular function because I am shown in the infinite series (oval modular function) using the exponential function of the ratio in two periods of the oval function.

The formula of the solution using the super geometric progression was shown by Kleine. For an outline, the solution of the regular icosahedron equation being shown in a super geometric progression and a regular icosahedron equation are led by チルンハウス conversion by being able to change into the general form of the fifth equation.

Numerical solution

Here, I speak elucidation by the numerical computation algorithm (basically the combination of infinite times of the addition, subtraction, multiplication and division operation). I assume the elucidation with the calculator, but cannot untie it with the calculator in a close meaning for a calculation to be able to originally play the current calculator in the limitedness time of operation and the Boolean operation with the integer ring because it is operation. However, I consider pseudoreal number expression called the floating point number or the true line expression of the complex number to be able to treat a complex number body than a possible thing. In addition, the repetition number of times before occupying the error range of the given positive price considers it that the operation in an infinity time is permitted substantially if there is the guarantee called the limitedness time. It is the approximate numerical solution in such a meaning.

Various technique is suggested, and the elucidation by the numerical computation algorithm continues the evolution now. Here, I write down some basic technique.

The elucidation by the Newton method gives you an initial value to become the candidate of the solution and considers a straight line to contact with the candidate of the solution to be the approximation of the original algebraic equation and is the method seeking the candidates of the next solution by untying the linear equation. I consider the candidate of the solution to be one of the solutions if judged to be settled within the error that the candidate of the solution gave this operation beforehand and I perform 減次 (deflation) and demand the next equation and put the Newton method again. I understand that I converge, and the second is early for a numerical solution (if I converge). But there are convergent badness for the equal root, being possible when I do not converge depending on an initial value, agony of the processing in case of the complex number, and there is little situation untying by the direct Newton method.

There is the elucidation called the base-up Suto method in the thing called handling of the complex number. This assumes operation to factorize the quadratic equation a concept.

Allied item

• Algebraic figure
• 冪根
• 1 冪根
• 求根 algorithm

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