A New Understanding on the Problem That the Quintic Equation Has No Radical Solutions

It is proved in this paper that Abel’s and Galois’s proofs that the quintic equations have no radical solutions are invalid. Due to Abel’s and Galois’s work about two hundred years ago, it was generally accepted that general quintic equations had no radical solutions. However, Tang Jianer et al. recently prove that there are radical solutions for some quintic equations with special forms. The theories of Abel and Galois cannot explain these results. On the other hand, Gauss et al. proved the fundamental theorem of algebra. The theorem declared that there were n solutions for the n degree equations, including the radical and non-radical solutions. The theories of Abel and Galois contradicted with the fundamental theorem of algebra. Due to the reasons above, the proofs of Abel and Galois should be re-examined and re-evaluated. The author carefully analyzed the Abel’s original paper and found some serious mistakes. In order to prove that the general solution of algebraic equation he proposed was effective for the cubic equation, Abel took the known solutions of cubic equation as a premise to calculate the parameters of his equation. Therefore, Abel’s proof is a logical circular argument and invalid. Besides, Abel confused the variables with the coefficients (constants) of algebraic equations. An expansion with 14 terms was written as 7 terms, 7 terms were missing. We prefer to consider Galois’s theory as a hypothesis rather than a proof. Based on that permutation group S 5 had no true normal subgroup, Galois concluded that the quintic equations had no radical solutions, but these two problems had no inevitable logic connection actually. In order to prove the effectiveness of radical extension group of automorphism mapping are proved to have no symmetries of Galois’s soluble group actually. It is invalid to use the solvable group theory to judge whether the high degree equation has a radical solution. The conclusion of this paper is that there is only the S n symmetry for the n degree algebraic equations. The symmetry of Galois’s solvable group does not exist. Mathematicians should get rid of the constraints of Abel and Galois’s theories, keep looking for the radical solutions of high degree equations.


Introduction
The so-called radical solution problem of quintic equation is to find a general formula to express the roots of arbitrary quintic equation in the radical forms of equation's coefficients uniformly.
People knew how to solve quadratic equations in the seventh century. Through the hard works of mathematicians, the general solutions of cubic and quartic equations were found about 450 years ago. But the general solutions of quintic equation and higher degree equations (called as high degree equations below) are still unknown at present, though some special solutions have been obtained for some quintic equations with special forms.
In the history of mathematics, Euler, d'Alembert, Lagrange, and Guass et al. tried to solve quintic equation but all of them failed. Until 1826, Abel published a paper to prove that general quintic equation cannot be solved [1]. Then Galois used the theory of group to prove the same result by proposing a judgment ruler for the solvability of general algebraic equations [2]. Galois's proof was regarded as more general and strict. After that, this problem was regarded to be perfectly solved. Most mathematicians stopped thinking about it.
Abel's paper proved that the general quintic equation had no solution, without saying that there were no radical solutions. Galois introduced the concept of radical expansion of domain and turned the problem into that the quintic equation had no radical solutions. There are subtle differences here, but Galois's saying often leads to misunderstandings. The absence of radical solutions for the quintic equation is a technical but narrow saying. More accurate one is that we cannot find a general formula to describe the solutions of a general quintic equation uniformly.
On the other hand, according to the fundamental theorem of algebra, there were n solutions for any n degree equation with one variable. It indicates that any n degree equation can always be written as Here 1 2 , , , n a a a Q ∈  are the rational numbers and 1 2 , , , n x x x  are the roots of the equation. If we add a constrain condition, suppose that Equation (1) is the so-called irreducible on Q, i.e., the equation has no the solutions of rational numbers, according to the fundamental theorem of algebra, it still has the solutions of irrational numbers or complex numbers.
However, according to the theories of Able and Galois, if ( ) f x is irreducible on Q, it means that the solutions 1 2 , , , n x x x  cannot be expressed by the radical sign forms of coefficient 1 2 , , , n a a a  , i.e., i x cannot be written in the forms of So the assertion that high degree equation has no radical solution is strange. If its solution cannot be expressed in the radical forms, what can we use to describe the solutions? As we known that number fields can be divided into rational number, irrational number, complex number and transcendental number. So-called having no radical solutions means that the roots of equations cannot be irrational numbers. Because we have defined that ( ) f x is irreducible on Q, the solutions of equations cannot be rational numbers. Because the transcendental numbers cannot be the roots of algebraic equations, what remained are complex numbers without containing radical signs.
On the other hand, it has been proved that the complex number roots of algebraic equations are conjugated. An odd order polynomial equation has a real number root at least [3]. The quintic equation has four imaginary number roots and one real number root, or two imaginary roots and three real roots. If these real roots are neither rational numbers nor irrational numbers or transcendental numbers, what are they?
It is obvious that the theory of Abel and Galois contradicts with the basic theorem of algebra. There must be one which is wrong. Gauss himself even proposed three or four methods to prove the basic theorem of algebra. After Gauss, many mathematician researched this problem. More than one hundred proofs were proposed up to now [4]. It notes that the basic theorem of algebra should be reliable, so the judgment that high degree equations have no radical solutions is doubtful.
Since the turn of this century, some progress has been made in the study of solving quintic equations. Professor Tang Jianer in Mathematics Department, Shanghai Finance and Economics University published a paper in 2012 to prove that same special quintic equations had radical solutions [5]. Other persons also made the same work [6] [7]. Zheng Liangfei solved a lot of quintic equations with number coefficients by using same special methods [8]. All of these solutions cannot be explained by the theories of Abel and Galois. Considering that the solution of quintic equation is one of the most fundamental problems in al-gebra, it is necessary to re-examine Abel and Galois's proofs.
The research in this paper finds that Abel's proof does not hold. His calculations contained logic confusion and basic conceptual mistake, which made Abel's paper hard to be understood. The general form of algebra equation's solution proposed by Abel is inconsistent with the practical forms of cubic and quartic equation's solutions.
Though the group theory of Galois was a great discovery, Galois's theory of solvable group had not solved the problem of high degree equation's solutions. We prefer to consider it as a hypothesis than to be a proof. Based on that the permutation group S 5 of quintic equation was a single group without true normal subgroup, Galois declared that the quintic equation had no radical solutions. Galois's deduction is ill-founded because both have no inevitable logic relation. The theory of radical extension did not match with the practical process of solving cubic and quartic equations too.
The main points of more detailed content are as follows. in Equation (1), the solution of quintic equation had the similar structure of cubic equation. These five equations calculated by Tang Jianer satisfy permutation group S 5 without true normal subgroups. There are no Galloi's solvable groups for them, but they still have radical solutions.
2) Abel's proof is analyzed carefully and some serious mistakes are founded. In order to prove that the solution form of general algebraic equation he proposed was valid for the cubic equation, Abel took the known solution of cubic equation as the premise to calculate the parameters of his equation. So Abel's proof is a circular argument without meaning.
3) Abel confused the concepts of variables with the coefficients (constants) of algebraic equations, violated the basic principle of solving equation. An expansion with 14 terms was written into 7 items, 7 items were missed. It is proved that the solutions of cubic and quartic equation do not satisfy the form given by Abel, and there is no reason to think that the solutions of higher order equation can satisfy the Abel's form. 4) Galois's solvable group theory is far-fetched and its conclusion and premise are contradictory. The Galois's theory admitted that the symmetries between the roots and the coefficients of quinic equations existed and used the permutation group S 5 to describe the symmetries. Due to that the S 5 group had no true normal subgroup, the quintic equation was regarded to have no radical solution. However, such an argument is not well-founded. Since it has assumed that the symmetry of the roots and coefficients can be described by the permutation group S 5 , it is actually accepted the existence of equation's solution. It just doesn't have the symmetry of true normal subgroup.
5) The automorphism mapping concept of radical extension was used to construct the Galois group. The function of an automorphism mapping group operator is to change the roots of an algebraic equation into other roots of the same However, in order to prove the effectiveness of this method for the cubic and quartic equations, Galois's theory actually used the algebraic relations of same roots to replace roots themselves. This is not only a substitution of concepts, but also introduces arbitrariness, results in the destruction of uniqueness.
6) The theory of radical extension did not match with the practical process. The practical process of solving an equation did not meet with the tower structure of

The Radical Solutions of Same Special Quintic Equations
Sheng Xinping earliest proposed the matrix method to solve the quartic equation [7]. Fang Jun and Kong Zhihui used the similar method to solve three kinds of special sextic equations by reducing the equations to cubic equations [4]. But this method cannot be used to solve the quintic equations.
Based on this method, Tang Jianer used five variables to replace one variable and obtained the radical solutions of several special quintic equations. This method has general significance [5]. The quintic equation was written in the standard form with 5 3 2 0 x px qx rx s The solutions of quintic cyclotomic equation 5 By introducing four variables , , , y z u v and using the fifth degree circular de- The relations between coefficients , , , p q r s and variables , , , y z u v are [5] ( ) According to Equation (2), Equation (4) indicates The five solutions of quintic equations are From Equations (12) and (13), we get 2 5 p r = . Form Equation (12), we have ( ) . Substituting them in Equation (14), we obtain The solution of Equation (15) is According to Equation (12), we get By considering Equation (14), it can be verified directly that Equation (19) is the solution of Equation (11). Another four solutions are determined by Equation (10). On the other side, for the cubic equation, we have 3 0 x px s + + = (20) It is interesting that the solution of quintic equation (11)  2) Let 0 p = in Equation (2), the equation becomes Therefore, one solution of Equation (22) is Another four solutions are determined by Equation (10). We can use a concrete example to verify the solution. For the equation We have 0 p = , 10 q = , 5 r = ,

Abel's Proof in 1824
The content in this chapter is cited from the first part of the Abel's original pa- Then he assumed that the solution of Equation (31) would be in following form Here m was a primer and 1 2 , , , , R p p p  had the same form as y. Continuously by this method, until they were represented as rational function of , , , , a b c d e .  , , , , m q q q q −  did not equal to zeros simultaneously, Equations (36) and (37) must have one or more than one common root. If the number of roots was k, we could find an equation with degree k. Its parameters were the rational function of

By using
From these k equations, the value of rational function z could be obtained which were be represented by 1 2 , , , , k r r r r  . Because these quantities themselves were the rational functions of , , , , , a b c d e R and 1 2 , , , p p p  , so z is the rational function of those quantities.
However, according to the definition , , , , m y y y y  represent the roots of m degree equation, α be the root of the equation Based on Equation (40), by considering the exchange symmetry of different solutions, as well as the result of a paper published by Cauchy in Journal de l'ecole polytechnic que [9], Abel declared that m could not be equal or greater than 5. In this way, Abel declared that the quintic equations had no solutions.

Abel's Supplement Proof in 1826
In Abel's paper in 1824, no concrete calculation was taken to prove the effectiveness of formulas (33)-(35). In his paper in 1826, Abel made the calculation to prove Equations (33)-(35). The proof is below [10].
For the cubic equation, take 0 c = for simplification, Equation (31) became   1  1  3  3  2  3  2  3  1 3  2 1 3  2  2  1  2   2  2  3  2  2 3 Therefore, for Equation (42), Abel thought that one also had It also indicated 0 p = , so Equation (42) Or writing Equation (49) as According to Equation (35), the results should be from Equation (52). Substituting it in Equation (53), the right side was just equal to zero. Substitute This is the second degree equation with two solutions There is the relation By considering Equations (56) and (57), Equation (48) can be written as Equations (58) and (43) had the same forms for Abel thought that his basic Formula (33) of general algebraic equation's solution was valid for the cubic equation.

The Problems Existing in Abel's Proof
In the Abel's proof, the concepts were unclear and logical was confused. The calculations had some serious mistake. For the quadratic equation, Equation But substituting Equation (42) in Equation (41), what we obtain is q p dp e q p d q p p p dp q pp q p pp q p q p Therefore, Equation (68) According to Equation (35), three equations are obtained ( ) Equations (71) In fact, in general situation, we have 0 On the other hand, let the four solutions of Equation (78) can be written as [11] (36) and (37) also have the problems. Equation (36) is only a definition, or an identical equation, in which both z and R are unknown. As we known, as a meaningful equation which can be solved, it should contain known quantity and unknown quantity. We use known quantity to represent unknown quantity. But (36) is not such equation.
Abel thought that Equations (36) and (37) had the common root. This is wrong. In fact, Equation (37)  In brief, because Equations (33) and (34) are untenable in general, the first part of Abel's proof in the paper in 1824 was wrong. The second part became meaningless. We do not need to discuss it any more.

The Symmetry between the Coefficients and the Roots of Algebraic Equations
General n degree algebraic equation can be written as ( ) Here the coefficients i a Q ∈ are rational numbers in general. According to the Gauss basic theorem, there are n roots 1 2 , , , n x x x  . So Equation (87) can be written as By expanding the multiplication of Equation (88) and comparing it with Equation (87), the relation between the roots and the coefficients are [12]   x ⇔ , its right side are unchanged. So there are symmetries between the roots and coefficients. In the words of permutation group, there exists S n symmetry for the algebraic equations.

The Galois's Theory of Solvable Group G n
By introducing the concept of radical domain extension, the symmetry of permutation group S n was transformed into the tower structure of Galois's group G n of radical expansion domain. Basic on it, the radical solutions of equations were discussed.
According to the Galois theorem, the necessary and sufficient condition that an algebraic equation has the radical solution is that that the corresponding Galois group is the solvable group. The definition of the solvable group is below [12]. Suppose that S n is the finite permutation group, there exist a series of normal subgroup which form the tower structure where I is the unit element, Because the permutation group S 5 corresponding quintic equation was a single group without true normal subgroup, the tower structure Equation (90) did not exist, Galois thought the quintic equation had no radical solution. Because S n group always has a subgroup S 5 when 5 n > , the higher degree equations were also considered to have no radical solutions.

The Galois's Theory of Radical Extension
In order to apply the solvable group theory to solve the algebraic equations, we need to introduce the concept of radical expansion [12]. It should be pointed out that Galois didn't really use the concept of radical expansion in his original paper, but there was such an idea. The strict radical extension theory and its most practical applications were the result of the improvement of Galois's early theory by later mathematicians. As a concept of number domain, the concept of expansion domain needs to meet some rules, but we do not go into details here.
What was discussed in the Galois's theory was the irreducible polynomial on as the extension of radical domain.
By solving the equation further, we obtain more roots, for example, we get , and so do. At last, all roots are founded, the whole root domain is described by According to the theory of Galois, if the radical extension can be carried out for an equation, the equation is considered to have radical solution. If the radical extension cannot be carried out, the equation is considered to have no radical solution. The quintic equation is considered to have no radical solutions because its radical expansion is considered impossible.

The Automorphism Mapping Group GalE/Q
After the concept of radical extension is introduced, in order to establish the relation between the radical extension and the permutation group, the automorphism mapping concept of radical extension is needed. Based on it, the automorphism mapping group GalE/Q is established. Then, through the Galois's correspondence theorem [13], GalE/Q was connected with the solvable group G n , or using G n to replace GalE/Q. By considering whether or not G n had the tower structure, Galois judged whether or not the equation had radical solution. So the automorphism mapping is the core concept of Galois's theory. It is the key to understanding the theory of Galois. We should make clear the logic relation here.
The operator σ is used to represent the automorphism mapping. It changes the root to another root of the equation or keeps the root self unchanged. Meanwhile, it does not change the coefficients of equation [12]. Suppose that i x is a root of equation (87), E is the root domain, we have ( ) 0 i f x = and define ( ) If the roots are known, by means of Equation (92), the concrete form of operator σ can be obtained. It is proved that the automorphism mapping forms group, called as the Galois group GalE/Q of radical extension.
By the complex reasoning process, it was proved that the automorphism mapping group GalE/Q was equivalent to the permutation group S n [12]. More complicated reasoning indicated that the automorphism mapping group was equivalent to the subgroup G n of permutation group S n (For example, the normal subgroup A 3 of S 3 ) [13]. We cannot cite them any more here, but can take some practical example. For the simplest quadratic equation  . It is obvious that the Galois group GalE/Q is equivalent to S 2 . There are four roots for the quartic equation To solve this problem, the so-called Galois corresponding relation was proposed [13]. It assumed that the Galois solvable group G n can be used to replace the radical extension group GalE/Q. Whether the G n group has the tower structure is equivalent to whether radical extension can be carried out.
Because the S 5 group has no real normal subgroup, G 5 is not the solvable group without the tower structure, the radical extension processes of quintic equations are considered impossible, so the quintic equations are considered to have no radical solutions.

The General Description of the Problem of Galois's Theory
It is pointed out that Galois only assumed that the quartic equation had no radical solution without really proving it. According to the definition, the automorphism mapping operator should be acted on the roots of equations. However, in order to prove that the theory was effective for the general cubic and quartic equations, some relations between the roots of equation had to be used to replace the roots themselves. This is violated the definition of automorphism mapping concept and led to the ineffectiveness of the proof. In fact, Galois only proved that the alternating group A n ( 5 n > ) was a single groups, but had not proved that the permutation group S n ( 5 n > ) was also a single groups. For example, S 5 is a true normal subgroup of S 9 , but it is not the only true normal subgroup. It is only an exception without true normal subgroup. All permutation groups S n which contain the non-prime cyclic groups as subgroups must not be single groups. How can we say that there are no radical solutions for all equations with 5 n > degrees?

Galois' Theory Cannot Explain Tan Jianer's Solution of Quintic Equation
It is obvious that Galois' theory cannot explain Tang

The Solutions of High Degree Equations with Number Coefficients
What the Galois's theory discussed was the equations irreducible on Q with arbitrary coefficients. However, there are great numbers of equations with number coefficients in practice. It is very difficult to judgment whether these equations are reducible or irreducible. At present, they are actually classified to the equations unsolvable. But it is not true, for example, for the quintic equation below  In fact, in many situations, the processes to judge whether or not an equation is reducible on Q is actually the processes to solve the equation. Therefore, it is actually meaningless to assume a high degree equation to be reducible on Q.
As we known that there is a theorem called the Eisenstein criterion which can be used to judge the reducibility of an equation. But it is only a criterion of sufficient condition, not a criterion necessary condition. Its application is limited.
For example, for Equation (96) Zheng Liangfei did a lot of work on the solutions of quartic equations with number coefficients and published a book titled "The solutions of Quantic Equations" [8]. Same effective methods were proposed to deal with various quantic equations with number coefficients. According to the opinion of Zheng Liangfei, any high degree equations with number coefficients can be solved in principle.

The Radical Solutions of High Degree Cyclotomic Equation
The equation with the form So k x are not rational numbers, for example, 1 1 2 It indicates that the radical solutions of (100) exist. However, the Vieta's formula of Equation (100) satisfies S 5 group which is a commutative cyclic group. According to the Galois's group theory, the cyclic group is not a solvable group without true subgroup [12]. But Equation (100) has radical solutions. Obviously, the Galois's theory is invalid for the quartic equation (100).

The Theory of Radical Extension of General Cubic Equation Is Invalid
The theory of radical extension is only a formal thing. In the actual processes to solve equations, the theory of radical tower extension is not obeyed. Although there is a large amount of literature to discuss the constructions of quadratic equations and some reducible cubic and quadric equations, only there is a little literature to discuss the radical tower constructions of general and irreducible cubic and quadric equations. The following is an example to construct the radical tower of general cubic equation cited from reference [12]. It can be seen that this proof is far-fetched and illogical. The general cubic equation is written as where , p q Q ∈ are rational numbers. The three solutions of Equation (102) are Here ( ) The group elements (123) and (132) are even permutations. According to the Galois's theory, the tower structure of solvable group is It is seen from the relation (109) that there are two steps in the process 3 G I → according to the solvable group theory. However, according the theory of radical extension, there are three additions of roots. The first extension is to add the first level root symbol with The second extension is to add the second level root symbol with ( ) ( ) The third extension is to add the imaginary number and root symbol ~3 i ω ( ) ( )   3  2  3  2  3 , , 3, Therefore, the tower structure of radical extension should be In order to make the practical process of radical extension satisfying the Galois's theory, a very complicated and full of holes proof was proposed in reference [12] 1) At first, Therefore, it is proved with had to be added in the base field to justify the theory Secondly, this method has not expressed how to obtain A 3 group through the automorphism mapping. The middle field B contains 2 3 , x x , but not contain x , then obtain another two. This is different from what described above. The Galois's theory of radical extension is invalid for the general cubic equation.

The Symmetry Change of Permutation Group in the Practical Radical Extension
According to Equations (103)-(105), the first solution of cubic equation is 1 x . The first radical extension is from rational number to irrational number. We have ( ) ( ) ( ) Then we obtain the second solution 2 x and the third solution 3 x . The second radical extension is from real number field to imaginary number field ( ) ( ) Thought two extensions, the right side of Equation (119)

It Is Illegal to Use the Relations between Roots Replacing the Roots to Construct Automorphism Mapping Operator
In common textbooks, a fatal mistake is made to prove the effectiveness of Galois's theory. In order to construct the radical extension groups, the automorphism mapping operators are acted on the relations between the roots of equations, rather than acted on the roots themselves. The method violates the definition of Equation (92) and is invalid. Unfortunately, this problem has always been ignored.
To illustrate the problem, the quartic equation is taken as an example. This example also shows that Galois's theory of radical extension is invalid for the quartic equation. Take a simplified quartic equation [14] ( ) 4 where , p q Q ∈ . Let 2 x y = , Equation (123) becomes the second degree equation 2 0 y py q + + =. Its two solutions are In order to prove that this process satisfied the Galois's theory, the relations The right sides of Equation (127) and (128) Then, introducing another relation  The relations between B and B 2 or B 3 are not that between the groups and its normal subgroups.
Because the radical extension theories of general cubic and quartic equations cannot match with the practical process, we have no reason to think that the X. C. Mei solvable group theory of Galois is effective for the quintic equation. For the high degree equations, the possible symmetry train of permutation group should be That is to say, the relation of roots and coefficients of an equation only has the symmetry of S n . The tower structure of Galois solvable group G n corresponding to the subgroup chain of S n group does not exist in the practical processes of solving equations.

The Resolvent of Cubic Equation
Lagrange introduced the concepts of permutation group and resolvent to solve the cubic and quadric equation in 1770 [12]. The effectiveness of Galois's theory of solvable group also needs to be verified by using it to construct the resolvent of lower degree equations.

The Problems Existing in the Galois's Theory to Explain the Resolvent of Cubic Equation
How to prove the validity of the resolvent of cubic equation by Galois's solvable group theory? The author finds that this is a vague problem in the textbooks and references.
The validity of Galois group on the preliminary solution type is involved here.
The permutation group of cubic equation is S 3 . Its unique true subgroup is A 3 shown in Equation (109), in which the group elements (123) and (132) 3  3  2  2  1  2  3  2  3  1   3  3  3  2  2  1  2  3  2  3  1   3  2  3  2  3  1 123  3  3  2  3  2  3  1  2  3  1  2   3  3  2  2  3  3  1  2  3  1  2 132 It is easy to see ( ) 3  , , x x x . Using 3 α and 3 β to replace 1 2 3 , , x x x confuses the concepts. According to the Galois's theory of solvable group, the automorphism mapping should be acted on 1 2 3 , , x x x , rather than acted on 3 α and 3 β . We cannot use the roots of the quadratic equation to construct the Galois group of cubic equation. Although it is valid using them to construct the roots of cubic equations, the validity of Galois' theory cannot be proved based on them.

The Resolvent of Quartic Equation
There were several versions using Galois's theory to construct the resolvent of quartic equation. For example, the versions proposed by Euler and Ferrari, but none of them can obtain the expected results in strict accordance with Galois's theory. The Euler's method is discussed below [14]. The quartic equation is Therefore, the conclusion of this paper is that Abel and Galois had not proved that the quintic equation had no radical solutions. Mathematicians should continue to work hard to find the general solutions of quintic and higher degree equations.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.