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.