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.