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.