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.