APPENDIX II ON ABEL'S PROOF OF THE THEOREM OF V., §11
Reading edition
Abel's proof (Œuvres, vol. 1, p. 545) is as follows[54]:
We have where is an irreducible polynomial of degree in . If we make use of the equation , we can introduce into this equation, and write it in the form where is a polynomial in the three variables , , and [55]; and we can suppose , like , of degree in and irreducible, that is to say not divisible by any polynomial of the same form which is not a constant multiple of or itself a constant.
From , we deduce and, eliminating , we obtain an equation of the form where and are polynomials in , , and . And in order that should be an integral of it is necessary and sufficient that
Abel now applies Lemma (2) of §11, or rather its analogue for polynomials in whose coefficients are polynomials in and , to the two polynomials and , and infers that all the roots of satisfy (3). From this he deduces that are all integrals of , and so that is an integral of . As (4) is a symmetric function of the roots of (2), it is a rational function of and , whence his conclusion follows[56].
It will be observed that the hypothesis that (2) does actually involve is essential, if we are to avoid the absurd conclusion that is necessarily a rational function of only. On the other hand it is not obvious how the presence of in affects the other steps in the argument.
The crucial inference is that which asserts that because the equations and , considered as equations in , have a root in common, and is irreducible, therefore is divisible by . This inference is invalid.
We could only apply the lemma in this way if the equation (3) were satisfied by one of the roots of (2) identically, that is to say for all values of and . But this is not the case. The equations are satisfied by the same value of only when and are connected by the equation (1).
Suppose, for example, that Then we may take Differentiating the equations and , and eliminating , we find Thus and these polynomials have a common factor only in virtue of the equation .
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