How large is the gap in Ruffini's 1813 proof that there is no general quintic formula?

Question

I'm reading Ruffini's final attempt at showing there is no general quintic formula which appeared in 1813 see here. (This is a much shorter proof than his first proof of 1799 in his Teoria generale delle equazioni... which proceeds via a lengthy analysis of the structure of the permutation group for five objects). It is commonly stated (e.g p. 499 of here) that Ruffini's proof contains a 'gap' which was filled in 1824 by part of Abel's proof that there is no quintic formula (see here), before Galois's writings around 1830. I notice on the other hand that the Wikipedia page on the Abel-Ruffini theorem (here) says that Ruffini's 1813 proof 'refined and completed' his earlier one. I'm interested in how much of a gap the 1813 proof still contains and would appreciate any comments on the below or answers to the question at the end.

If we denote a permutation of the five variables $x_1,\ldots x_5$ by, for example, $(12345)$, meaning $x_1$ is replaced with $x_2$, $x_2$ is replaced with $x_3$ etc., then Ruffini's proof is essentially:

(a) First show that if $\Pi$ and $y$ are functions of the five variables $x_1,\ldots x_5$ such that $y^p=\Pi$ and if the value of $\Pi$ is unchanged under the cyclic permutations $(12345)$, $(123)$ and $(345)$, then the value $y$ is also unchanged under these permutations. To show this, let $y_1$ denote the result of applying the permutation $(12345)$ to $y$, then $y_1=\beta y$ for some $\beta$ as they are both roots of $y^p=\Pi$. This permutation has order five and so $\beta^5=1$. Let $y_a$ denote the result of applying the permutation $(123)$ to $y$ then in similar way $y_a=\gamma y$ for some $\gamma$ with $\gamma^3=1$. Next let $y_b$ denote the result of applying the permutation $(123)$ and then $(12345)$ to $y$, so $y_b=\beta y_a=\beta\gamma y$ and as the combined permutation has order 5, $y=(\beta\gamma)^5 y$ and so $\beta^5\gamma^5=1$. As $\beta^5=1$, it follows that $\gamma^5=1$ and as $\gamma^3=1$, it also follows that $\gamma^2=1$. From $\gamma^3=1$ and $\gamma^2=1$ we get $\gamma=1$. Now let $y_c$ be the result of applying the permutation $(345)$ to $y$, then $y_c=\delta y$ and the same reasoning as above gives $\delta=1$. Finally, noting that $(123)$ followed by $(345)$ gives $(12345)$ shows that $\beta=\gamma\delta$ and so $\beta=1$. In summary, $\beta=\gamma=\delta=1$ and so the value of $y$ is unchanged by the permutations $(12345)$, $(123)$ and $(345)$. (See p. 334 of here for a more formal presentation of this argument).

(b) Next consider a quintic equation $x^5+Ax^4+\ldots+E=0$ with roots $x_1,\ldots,x_5$, supposing that there is a formula to find the roots from the coefficents by a series of rational functions and radicals. First let $P',P'',P''',\ldots$ be rational functions of $A, B, C, \ldots$. Then take roots of any of these, e.g. let $Q'=\alpha'\sqrt[p']{P'}, Q''=\alpha''\sqrt[p'']{P''}, \ldots$ where $p',p'',\ldots$ are integers and $\alpha',\alpha'',\ldots$ are $p'$-th, $p''$-th etc. roots of unity. Now let $R', R'',\ldots$ be roots of rational functions of the corresponding $P'$s and $Q'$'s, i.e. $R'=\beta'\sqrt[q']{F'(P',Q')}, R''=\beta''\sqrt[q'']{F''(P'',Q'')}, \ldots$. In a similar manner form $S', S'', \ldots$ as roots of functions of the corresponding $P, Q$ and $R$ etc.. In this way we can express a root of the quintic in the form: $$\tag{1}x_1=F(P',P'',\ldots,Q',Q'',\ldots,R',R'',\ldots,S',S'',\ldots,\ldots)$$

(c) As $P',P'',\ldots Q',Q''\ldots$ all ultimately depend on the coefficients $A, B, \ldots$, this formula can be written in terms of the coefficients. Now $A, B, \ldots$ are (symmetric) functions of the roots, i.e. $A=-(x_1+x_2+\ldots), B=x_1x_2+x_1x_3+\ldots$ and so we can rewrite the right hand side of (1) in terms of $x_1,x_2\ldots$

(d) As the expressions for $A, B, \ldots$ are symmetric functions of $x_1,x_2,\ldots$, the values of $A, B, \ldots$ are unchanged by any permutation of $x_1,x_2, \ldots$. In particular they are unchanged by the permutations of the roots represented by $(12345)$, $(123)$ and $(345)$. Similarly, $P', P'', \ldots$ are rational functions of $A, B, \ldots$ and so unchanged by these three permutations. By the reasonining in (a), $Q',Q'', \ldots$, which are roots of $P',P'',\ldots$ are unchanged by these permutations, and so $F'(P',Q')$, $F''(P'',Q'')$ etc. are unchanged by the permutations. Again, by the reasoning in (a), $R', R'', \ldots$ are unchanged by the three permutations, as are $S', S'',\ldots$ etc. As the right hand side of (1) is a rational function of $P',P'',\ldots,Q',Q'',\ldots,R',R'',\ldots,S',S'',\ldots,\ldots$, this is unchanged by the permutations of roots represented by $(12345)$, $(123)$ and $(345)$.

(e) We now have the contradiction that the right hand side of (1) is unchanged by the permutations, but the left hand side, i.e. $x_1$, is changed by them, e.g. $x_1$ changes to $x_2$ under $(123$). This contradiction shows that there cannot be a formula for the roots of a quintic in term of radicals starting from the coefficients of the quintic.

As mentioned, the general consensus seems to be that (a) is valid, but that (b)-(e) contain a gap that was completed by Abel in 1824, Abel showing that if a polynomial can be solved by radicals, then any of the radicals contained in the solution can be expressed as rational functions of the roots of the polynomial. An example of what this means is to substitute $b=-(x_1+x_2)$ and $c=x_1x_2$ into the square root in the quadratic formula for $x^2+bx+c=0$, i.e. $\sqrt{b^2-4c}$, giving $\sqrt{x_1^2+x_2^2-2x_1x_2}=\sqrt{(x_1-x_2)^2}$ which is either $x_1-x_2$ or $x_2-x_1$, i.e. a rational function of the roots. The reason often given for why this matters is that expressing the roots of polynomials often involves going outside of the field containing the coefficients of the polynomial and the roots. For example, to express the roots of $x^3-15x-20=0$ in exact form requires the use of complex numbers even though all three roots are real (as can be seen by solving the cubic in Wolfram Alpha). Abel's proof fills this gap by showing that even if we go outside of the field containing the coefficients and roots when writing an expression for the root, each radical in the expression is still in this field, i.e. all radicals are like the square root in the quadractic formula just mentioned. Michael Rosen also expresses this here as `If $L$ is contained in a radical tower over $K$, then $L/K$ is itself a radical tower' - see Step 1 on page 499 which Rosen proves using an argument based on Abel's.

Despite understanding (I believe) what Abel shows, I'm still not clear as to why this step is necessary to complete Ruffini's proof. It seems to me that, the main result of (a) above applies to any function of five roots, not just rational functions. All of the $P',P'',P''',\ldots$, $Q',Q'',Q''',\ldots$, $R',R'',R''',\ldots$ etc. can be written as functions of the five roots as they're all based on substituting the symmetric functions of the roots for $A, B, \ldots$. They may contain radicals but it's shown at each step that these are not multi-valued as so can't contribute to any multi-valuedness of $R',R'',R''',\ldots$ etc.

For example, suppose part of a formula for solving a quintic took the form $\root{3}\of{M+\sqrt{N}}$ where $M$ and $N$ are functions of the coefficients $A, B, \dots$, rather like in Cardano's solution to the cubic. Ruffini's argument shows that the inner $\sqrt{N}$ is single-valued, and so $M+\sqrt{N}$ is a function of $A, B, \ldots$ and a single valued term, and so as far as I can see we can apply Ruffini's argument in (a) to the cube root, showing that it only takes one value.

So, my question: is Abel's step indeed needed to complete Ruffini's proof and if so can someone explain what is wrong in my reasoning in the last two paragraphs? As a secondary, perhaps related question, Ruffini's argument is also criticised here (on p. 344) for stating that the format of any solution can be put into the form shown in (1) above - is this a valid criticism?

Answer 1

The result by Abel (also called theorem on natural irrationalities) is very important in the proof of insolvability of a general quintic. It ensures that at any stage of forming radicals over field of coefficients we do not go beyond the field of roots of a polynomial.

What would happen if this wasn't ensured? Let us take a simple case of a general quadratic polynomial $x^2+ax+b$ with roots $x_1,x_2$ and let us note the following formula for roots $$x_1=\frac{-a+\sqrt{a^2-4b}}{2}$$ If the radical $\sqrt{a^2-4b}$ is not multivalued and also not a rational function of the roots then its value is unchanged when we permute $x_1$ and $x_2$ and formula above changes to $$x_2=\frac{-a+\sqrt{a^2-4b}}{2}$$ which means that $x_1=x_2$ which is not the case for a general quadratic polynomial.

On the other hand if we assume that radical $\sqrt{a^2-4b}$ is a rational function of roots as well as having a single value (eg $x_1-x_2$) then the formula for roots becomes $$x_1=\frac{(x_1+x_2)+(x_1-x_2)}{2}$$ which remains valid under the transposition $(12)$.

This is one of the reasons that a general cubic polynomial $x^3+ax+b$ can't be solved by radicals over the field of coefficients $\mathbb{Q} (a, b) $. The formula by Cardano for roots does not involve any complex numbers but the radicals in the formula can't be expressed as rational function of roots with rational numbers as coefficients and hence one can't get all the roots by applying permutations. If we work in the field $\mathbb{C} (a, b) $ then we can express the radicals as rational function of roots with complex coefficients and the issue is resolved.

The argument in part a) of your question is a crucial ingredient of the proof and it will not work unless we assume the functions to be rational functions of roots. The multivalued nature of radicals is important in order to get all the roots otherwise any radical function made by the coefficients of a polynomial will always remain invariant (just as the coefficients themselves) under any permutation of the roots.

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Based on response from asker via comments (to this answer) let me emphasize further on the role played by radicals in the solution of polynomial equations. It is expected that there is a formula for root of a polynomial consisting of nested radicals and involving coefficients of the polynomial. The radicals involved must be treated as multi valued functions of their arguments and by using a suitably chosen combination of the values of radicals one can get all the roots of the polynomial.

When this is the case then it is very difficult to make sense of permutations of roots applied to these radical expressions. It is not at all clear which permutations will lead to which particular values of the radical. A more precise way was put forth by Abel and he proved that these radicals can be expressed as rational functions of roots with complex coefficients (we need the existence of roots of unity in an essential manner here) and permutation of roots will lead to different values of the radical expression.

Let us now understand the argument in part a) in language of automorphisms. A permutation of roots applied to a rational function of the roots is essentially an automorphism on the field of such rational expressions. If $y, \Pi$ are two such rational expressions of the roots and $p$ a prime number such that $y^p=\Pi$ then we can apply a permutation $\sigma $ on this equation to get $\sigma(y^p) =\sigma(\Pi) $ which further leads to $(\sigma(y)) ^p=\sigma(\Pi) $ and $(\sigma(y) /y) ^p=1$ (provided $\Pi$ is invariant under $\sigma$). Now one can put forth the argument by Ruffini in a precise and unambiguous manner and get the desired result (part (a) of your question).

The key aspect of the Ruffini's proof is that making radicals of rational expressions in various indeterminates in general destroys the symmetries (invariance under permutations of the indeterminates), but if there are too many indeterminates involved (leading to too many symmetries) then some symmetries are preserved no matter how many times we make radicals and therefore the process can not lead to the roots which are totally non-symmetric.

Galois described this aspect by saying that the process of forming radicals either leaves the Galois group unchanged or partitions it into a number of smaller groups in a very specific way (the word group and its partitioning as used by Galois in his writings has a different meaning compared to the the modern meaning, but it corresponds to the way a Galois group can be partitioned into cosets via a normal subgroup). In this manner the symmetries are reduced on forming radicals.

The key idea of the proof for insolvability of the general quintic equation via radicals was given by Ruffini and he made an assumption regarding nature of radicals (probably thinking it as obvious or not needing a proof). This is something which Abel noticed and justified and he provided another proof based on a theorem of Cauchy regarding number of values taken by an expression of several variables under the permutations of the variables. This proof by Abel is not presented in a coherent manner and I haven't grasped it fully so far. Compared to this the proof by Ruffini (and its later simplification by Wantzel) based on invariance of certain expression under the permutations $(123),(345)$ is really easier to comprehend and in my opinion is rather remarkable.