Section 2: Properties of algebraic functions which satisfy a given equation: Part 2

Translated by me as: 

Now I say that equation (3) can not take place unless there is separately

r₀  =  0 , r₁ =  0  . . . r_n-1 =  0 .

Indeed otherwise we would have with  p ^{1 / n}  =  z , [original text corrected]  the two equations

z ⁿ  -  p =  0 , and

r₀ +  z ¹  + r₂  . z ²   ...+  r _n-1 . z ^n-1 =  0

Commentary: 

z satisfies an equation of nth order and also an equation of order n-1

Section 2: Properties of algebraic functions which satisfy a given equation: Part 1

Translated by me as:

                                                                                  II

                                   Properties of algebraic functions which satify a given equation.

Let
                                                 c₀ + c₁ y + c₂ y ² ... c_r-1 y ^r-1 + c_r y ^r = 0.   . . . . . . .  1.

be an equation of degree r, where  c₀ , c₁ ... are rational functions of x', x'' ...,  x', x'' ... being independent of any quantities. Suppose we can satisfy this equation by putting in place of  y an algebraic function of x', x'' ...
Let

                                          y =   q₀ +  p ^{1 / n} + q₂  . p ^{2 / n} + ... q _n-1. p ^{( n-1) / n}  . . . . . . . . . .  2.

be this function. Substituting this expression for y in the proposed equation, we obtain , by virtue of the forgoing, an expression of the form.

                                         r₀ +  p ^{1 / n} + r₂  . p ^{2 / n}  ...+ r _n-1 . p ^{( n-1) / n} = 0.  . . . . . . . . 3.

 where  r₀ , r₁ , r₂ ,.., r_n-1 are rational functions of  p , q₀ , q₂ ... q_n-1. [original text corrected]


Commentary:

Once again there is a typo where q₁ should be q₂ since q₁ has previously been set to 1. Here y is a general algebraic function so based on the preceding sections we could say that y an algebraic function of order μ and degree m.

                                  y =   q₀ +  p ^{1 / n} + q₂  . p ^{2 / n} + ... q _n-1. p ^{( n-1) / n} ; 

where n is a prime number, q_0, q_{2 ...} q _n-1 are algebraic functions of order μ and degree m - 1 and also, p is an algebraic function of order μ - 1 and unrestricted degree and  p ^{1 / n} cannot be expressed as a rational function of  q_0, q_{1 ...} q _n-1.   Abel does not address the issue that y is a multi-valued function. Presumably not all the possible values of y are expected to be solutions of the given equation. In the case of the well know solution to the quadratic equation both values of the square root do yield solutions of the given equation. In the case of the solution of the cubic equation the picture is not so clear. The algebraic function which is a solution of a given cubic equation will have multiple terms with square roots under cube roots. It seems more than 3 different values would result yet we know the cubic equation cannot have more than 3 solutions. 

Section 1: General expression of an algebraic function: Part 11

Translated by me as:

From all the forgoing  we conclude:
If v is an algebraic function of order μ and degree m, we can always pose:

                                  v =   q₀ +  p ^{1 / n} + q₂  . p ^{2 / n} + ... q _n-1. p ^{( n-1) / n} ; 

where n is a prime number, q_0, q_{2 ...} q _n-1 are algebraic functions of order μ and degree m - 1 and also, p is an algebraic function of order μ - 1 and  p ^{1 / n} cannot be expressed as a rational function of  q_0, q_{2 ...} q _n-1. [original text corrected]

Commentary:
 
This is just a restatement of what has been established up to this point. There is a misprint where q₁ should be replaced by q_2. Abel does not mention it but p is of unrestricted degree.

Section 1. General expression of an algebraic function: Part 10

Translated by me as:

In the expression above for v, we can always have q₁ = 1. For if q₁ is not zero, we obtain it by  p₁ = p . q₁ ⁿ

                                                    p =    p₁  /   q₁ ⁿ    and        p ^{1 / n}  =     p₁ ^{1 / n} /  q₁        , so

                                                                                                             
                                           v =   q₀ +  p₁ ^{1 / n} + (q₂ / q₁²) . p₁ ^{2 / n} + ... (q _n-1 / q₁^{( n-1)} ). p₁ ^{( n-1) / n}   

an expression of the same form as the preceding only that q₁ =  1. If  q₁ =   0, let  q _μ  be one of the quantities  q_0, q_1...q _n-1 that is not zero and let   q _μ ⁿ_.  p ^μ ₌  p_{1 .}  We conclude that  q _μ ^α_.  p ^{α μ / n} ₌  p₁^{α / n} . So taking two integers α and β, which satisfy αμ - βn = μ', μ' being an integer, we will have

                                                                  
                                                 q _μ ^α_.  p ^{( β n+ μ' ) / n}  =  p₁^{α / n}  and   p  ^{μ'  / n} = q _μ ^{- α}_.  p ^{- β}_.  p₁^{α / n}  .  

After all this and noticing that  q _{μ  .}  p  ^{μ / n} ₌  p₁ ^{1 / n}   _, v has the form

                                 v =   q₀ +  p₁ ^{1 / n} + q₂  . p₁ ^{2 / n} + ... q _n-1. p₁ ^{( n-1) / n} 
 
Commentary:

Abel shows that we can always express v in the previously established form where  q₁  can be made to be 1. He does not explain why this is necessary or even desirable but it seems important to him to show this. Abel also makes use of the restriction that n is a prime number.

Section 1. General expression of an algebraic function: Part 9

Translated by me as:

Let μ be any integer, we can always write

                                                                        μ = an + α

a and α are integers and  α < n.  It follows that


                                                              p ^{μ / n}  =  p ^{( a n + α ) / n}  =  p ^a . p ^{α / n}

so putting this expression instead of   p ^{μ / n} in the expression of v, we obtain,

                                                      v =   q₀ + q₁ p ^{1 / n} + q₂ p ^{2 / n} + ... q_n-1 p ^{( n-1 ) / n} 

 q_0, q_1, q₂ still being rational functions of  p, r', r''... and consequently functions of order μ and degree (m-1) and it can be assured that these quantities are related so that it is impossible to express p ^{1 / n}  rationally in these quantities.

Commentary:

Abel shows that his general form need only to include powers of  p ^{1 / n}  up to (n-1). At this point q_0, q_1, q₂ are not the same as the old q_0, q_1, q_2.

Section 1. General expression of an algebraic function: Part 8

Translated by me as:

Let  V_1, V_{2 ...} V_n-1 be the n-1 values of V, found successively using αp^1/n , α²p^1/n , α³p^1/n ...α^n-1p^1/n in place of  p^1/n , α being a root different from unity of the equation  α ⁿ - 1 = 0;  By multiplying the fraction T/V above and below by V_1. V_2. V_3... V_n-1

                                                                         T.V₁.V_2... V_n-1
                                                                v =  —————————
                                                                         V.V₁.V_2... V_n-1

The product  V.V_1... V_n-1 may as we know , be expressed as an entire function of  p and the quantities r', r''...,  and the product T.V_1... V_n-1 is, as we see, an entire function of  ⁿ√ p and of  r', r''... By posing the product equal to

                                                              s₀ + s₁ p ^{1 / n} + s₂ p ^{2 / n} + ... s_k p ^{k / n} 

we find

                                                                         s₀ + s₁ p ^{1 / n} + s₂ p ^{2 / n} + ... s_k p ^{k / n}
                                                                v =  ——————————————————
                                                                                                m

or by writing  q_0. q_1. q_2... instead of s₀ /m, s₁ /m, s₂ /m etc. 

                                                              v =   q₀ + q₁ p ^{1 / n} + q₂ p ^{2 / n} + ... q_k p ^{k / n} 

where q_0, q_{1 ...} q_k are rational functions of the quantities p, r', r'' etc.

Commentary:

Abel further simplifies the dependence on the term ⁿ√ p. Abel does not mention that the entire function produced by V.V_1... V_n-1 is expressed by the term m. This m is therefore different from the m that he has previously introduced. Also Abel does not support his assertion that m is an entire function of  r', r''... and that T.V_1... V_n-1 is an  entire function of  ⁿ√ p and of  r', r''... These assertions are well explained in Pesic Appendix B except Peter claims that the terms q_0, q_{1 ...} q _k would be of order μ-1 at most, while Abel has made it clear that they would be of order μ at most.

Section 1. General expression of an algebraic function: Part 7

Translated by me as:

In the forgoing we have seen, a rational function of several individual quantities can always be reduced to the form

                                                                                    s
                                                                                    t

where s and t are entire functions of the same variable quantities. It is concluded from this that v can always be expressed as follows
                                                                          φ(r', r''...ⁿ√ p)
                                                                v =  ————————
                                                                         τ(r', r''...ⁿ√ p)

where φ and τ are entire functions of r', r''... and ⁿ√ p. Based on the above we have found that  any entire function of several quantities s, r', r'' ... can be expressed by the form

                                                                   t₀ + t₁ s + t₂ s ² + ... t_m s ^m

t_{0 ,} t_{1 ...} t_m  being entire functions of r', r'', r'''... without s. We can therefore let

                                                             t₀ + t₁ p ^{1 / n} + t₂ p ^{2 / n} + ... t_m p ^{m / n}                T

                                                  v =  ————————————————————  =   ——
                                                             v₀ + v₁ p ^{1 / n} + v₂ p ^{2 / n} + ... v_m' p ^{m' / n                  V}


where  t_{0 ,} t_{1 ...} t_m   and  v_{0 ,} v_{1 ...} v_m'  are entire functions of r', r'', r''', etc.

Commentary:

Abel expresses  algebraic function in a way which emphasizes it's dependence on one of it's higher order quantities  ⁿ√ p. This expression is very general and perhaps the only restriction Abel places on it is that the quantity ⁿ√ p is one of it's highest order terms, which means that it should be at least tied in terms of having the greatest number of nested radicals. The m introduced here is different from the m previously introduced to represent the degree of the algebraic function

Section 1. General expression of an algebraic fuunction: Part 6

Translated by me as:

From which it follows, we can say

                                                                   v = f(r', r''...ⁿ√ p)

where p is a function of  μ-1 order but r',r''... are functions of  μ order and at most m-1 degree, and we can always assume it is impossible to express ⁿ√ p as a rational function of these quantities.


Commentary:


Here v is from the previous discussion an algebraic function of order μ and degree m. Abel chooses to emphasize the dependence of v on one particular radical term and express the dependence on all other radical terms by using the quantities  r',r''....

Section 1. General expression of an algebraic function: Part 5

Translated by me as:

If in the expression of v the number of quantities ^n'√ p', ^n''√ p''... is equal to m, we say, that the function is of the μ^th order and the  m^th degree. We see then that a function of order μ and degree 0 is the same as a function of order μ-1 and a function of order 0 is the same as a rational function.


Commentary:


Abel refines the hierarchy of the algebraic functions to include degree, which is a sub-classification of order.

Section 1. General expression of an algebraic function: Part 4

Translated by me as:

We can evidently assume that it is impossible to express the quantities ^n'√ p', ^n''√ p''...  by a rational function of the other quantities r', r'' ...; because otherwise the function v has a simpler form. 

                                                                    v = f(r', r''...^n'√ p', ^n''√ p''...)

where the number of quantities ^n'√ p', ^n''√ p''... would be reduced by one unit. By reducing in this manner the expression of v whenever possible there would be an expression which is irreducible or an expression of the form

                                                                    v = f(r', r'', r''' ...) 


but this function is only of order μ-1 which is a contradiction.


Commentary:


Abel argues that he can assume that the function is not reducible to an expression of a lower order. The argument is that if it were reducible then we would reduce it before preceding.

Section 1. General expression of an algebraic function: Part 3

 

Translated by me as:

So in designating μ as the order of the algebraic function v

                                                    v = f(r', r''...^n'√ p', ^n''√ p''...)

where  p', p'' are functions of order μ-1; r', r'' ... are functions of order μ-1 or lower and n', n'' ... are prime integers. f always signifies a rational function of the quantities between the brackets. 

Commentary:

Abel shows how to determine the order of an algebraic function 

Section 1. General expression of an algebraic function: Part 2

Translated by me as:

Continuing in this manner we obtain the algebraic functions of the third, the fourth ... the μ^th order and it is clear that the expressions of functions of   the μ^th order will be an expression of  general algebraic functions.

Commentary:

Abel claims that his form is a general form of an algebraic function.

Section 1. General expression of an algebraic function: Part 1

Translated by me as:

We will now find the general form of algebraic functions. Denote by f(x', x''...) any rational function whatsoever. It is clear that every algebraic function can be made using the operations designated by f(x', x''...)  combined by the operation    ^m√ r where m is a prime integer. So, if p', p''... are rational functions of x', x''... then

                                                                p₁ = f(x', x''...^n'√ p', ^n''√ p''...)

is the general form of the algebraic function of x', x''... in which the    ^m√ r affects only rational functions. The functions of the form  p₁ will be called first order algebraic functions. We designate by p₁', p₁'' multiple quantities of the same form as  p₁ . The expression

                                                p₂ = f(x', x''...^n'√ p', ^n''√ p''... ^n₁'√ p₁', ^n₁''√ p₁'' ...)

is the general form of the algebraic function of x', x''... in which the    ^m√ r affects only rational functions and algebraic functions of the first order. Functions of the form p₂ will be called second order algebraic functions. In the same manner the expression

                                  p₃ = f(x', x''...^n'√ p', ^n''√ p''... ^n₁'√ p₁', ^n₁''√ p₁'' ...^n₂'√ p₂', ^n₂''√ p₂'' ...)

in which p₂', p₂'' are algebraic functions of the second order, is a general expression of an algebraic function in  x', x''... in which the    ^m√ r affects only rational functions and first and second order algebraic functions.

Commentary: 

Abel develops the concept of the order of an algebraic function and then expresses the general form of an algebraic function up to the third order. We can see that in general these algebraic functions will be multi-valued.

Section 1. General expression for a rational function

Translated by me as:

Now consider the rational functions. When f(x', x'' ...) and φ(x', x'' ...) are two entire functions, it is evident that the quotient

                                                                              f(x', x'' ...)
                                                                             φ(x', x'' ...)

is a special case of the result of the first three operations, which are rational functions. It can therefore be considered a rational function as the result of the repetition of these operations. If we denote by v', v'', v''' etc several functions of the form


                                                                            f(x', x'' ...)
                                                                           φ(x', x'' ...)


we can easily see that the function

                                                                            f(v', v'' ...)
                                                                           φ(v', v'' ...)


can be reduced to the same form. It follows that any rational function of several quantities x', x'' ... can always be reduced to the form

                                                                              f(x', x'' ...)
                                                                             φ(x', x'' ...)


where the numerator and denominator are entire functions.


Commentary:


Abel derives a general expression for a rational function. For him this result is easy so his explanation is not detailed. His idea seems to be that whenever two expressions of this form are combined with any of the three allowed operations it results in an expression of this same form. So no matter how many times or in what combinations you use the three allowed operations you will always be left with an expression of this same form.

Section 1. General expression for an entire function

Translated by me as:

Let  f(x',x'',x'''...)  be any function that can be expressed as a finite number of terms of the form

                                                                 Ax' ^m₁ . x'' ^m₂ ......

where A is a quantity independent  of x', x'' etc. and m₁, m₂ etc are positive integers; it is clear that the operation designated by  f(x',x'',x'''...) is a special case of the first two steps above. We can therefore consider the entire functions according to their definition as resulting from a limited number of repetitions of this operation. Thus by appointing v', v'', v''' etc. as several functions of  x', x'', x''' ...., of the same form as f(x',x''...) the function f(v', v'' ...) will evidently be of the same form as f(x',x''...). Now f(v', v'' ...) is the general expression of functions resulting from the operation f(x',x''...) twice repeated. We therefore find the same result by repeating this as many times as you like. It follows that any entire function of several quantities x', x'', x''' ..... can be expressed by a sum of several terms of the form Ax' ^m₁ . x'' ^m₂ .....

Commentary:

Abel derives a general expression for any entire function.

Section 1. Definition of rational and entire functions

Translated by me as:

When the function v can be formed by the first three operations above, it is called rational; and if the first two operations only are necessary, it is called rational and entire, or just entire.

Commentary:

Abel defines what he means by a rational function and an entire function.

Section 1. On the general form of algebraic functions

Translated by me as:

On the general form of algebraic functions.

Let x', x'', x''' ... be a finite number of arbitrary quantities. We say that v is an algebraic function of these quantities, if it is possible to express v using x', x'', x''' ... in the following steps. 1. by addition; 2. either multiplying by quantities dependent on x', x'', x''' ... or by quantities not dependent; 3. by division; 4. by the extraction of roots with prime exponents. Among these operations we have not included subtraction, elevation to integer powers and the extraction of roots with compound exponents because they are obviously included in the four operations mentioned.

Commentary:

Abel defines what it is that he means by the term algebraic function. Right at this initial step a subtlety has snuck in the room. Extraction of roots will be an allowed operation but extraction of roots is known to produce multiple values. We will put this observation aside but we should occasionally ask the question whether an expression involving a root extraction implies all possible extracted values or something less than all possible extracted values.

Opening Paragraph

Translated by me as:

One can, as we know, solve the general equation of the fourth degree, but the equations of a higher level only in special cases, and if I am not mistaken, no one has satisfactorily answered the question:

"Is it possible to solve algebraic equations in general , which exceed the fourth degree?"

This thesis aims to answer this question. Algebraically solving an equation does not mean anything other than to express it's roots by algebraic functions of the coefficients. First we must consider the form of algebraic functions, and then look if it is possible to satisfy a given equation by putting the expression of an algebraic function instead of the unknown.

Commentary:
Able outlines the approach that he will take to ascend Mount Impossible

Abel's 1826 Impossibility Proof

Translated by me as:
"Demonstration of the impossibility of the algebraic solution for general equations greater than the fourth degree."

Commentary:
So begins Abel's 1826 paper published in Paris on the impossibility of solving the general quintic equation in an algebraic form. I haven't been able to find a translation for this important work so I'm going to be providing it in this blog along with commentary on Abel's proof. I found the pdf for this French paper at:
http://www.abelprisen.no/verker/oeuvres_1839/oeuvres_completes_de_abel_1_kap02_opt.pdf

My objective here is to follow Abel's advise to "learn from the masters, not from their students"