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