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.