Abel’s theory of equations solvable by radicals, Abelian Transformations

Abel’s proof that the general quintic cannot be solved by radicals is sometimes presented as though it ended the classical theory of algebraic equations. In fact, it posed a new and more precise problem. Once one knows that no universal radical formula can solve an arbitrary equation of degree five, the natural question is no longer whether a sufficiently ingenious formula has escaped discovery. The question becomes structural: what special feature of an equation makes radical solution possible? Abel returned to this question in his later memoir on a particular class of equations solvable algebraically. The earlier theorem had been negative: most equations are too complicated to yield to radicals. The later memoir is constructive: it identifies a large and natural mechanism by which an equation of any degree can be reduced and solved.

The word “algebraically” in Abel’s title means by rational operations and extraction of roots. Starting from known quantities, one is allowed to add, subtract, multiply, divide, and take roots of any order. Abel does not begin with a formula in coefficients. Instead he begins with a root and asks how the other roots can be obtained from it. His central idea is that an equation becomes tractable when its roots are connected by rational transformations that fit together compatibly. A single rational transformation can split the roots into cycles and lower the degree of the problem. A cyclic transformation can be diagonalized by root-of-unity sums and solved explicitly by radicals. Finally, if all the rational transformations among the roots commute, then the first reduction reproduces the same favorable structure in a smaller equation. The reduction can then be repeated until nothing remains to solve.

Let $\Phi(X)=0$ be an irreducible equation of degree $\mu,$ with coefficients among a fixed stock of known quantities. Let $x$ be one of its roots. Abel assumes that another root can be obtained from $x$ by a rational expression. Thus there is a rational function $\theta$ , with known coefficients, such that

$\displaystyle x'=\theta(x)$

is also a root of $\Phi(X)=0$ . At first this looks like only a relation between two particular roots. Abel’s first observation is that irreducibility turns it into a relation among the entire collection of roots. The elementary principle Abel uses is the following. Suppose that the irreducible polynomial $\Phi(X)$ has one root $x$ , and suppose that another polynomial $F(X)$ , built from known coefficients, vanishes at $x$ . Then every root of $\Phi$ also vanishes under $F$ . Indeed, divide $F$ by $\Phi$ :

$\displaystyle F(X)=Q(X)\Phi(X)+R(X),\quad \deg R<\mu.$

At $X=x$ , this gives $0=F(x)=R(x).$ If $R$ were not the zero polynomial, then $x$ would satisfy an equation of degree strictly smaller than $\mu$ , contradicting irreducibility. Hence $R=0$ , so $\Phi$ divides $F$ . Every root of $\Phi$ is therefore a root of $F$ .

This is now applied to the rational transformation $\theta$ . The statement $\Phi(\theta(x))=0$ can be cleared of denominators, producing a polynomial relation in $x$ with known coefficients. Since it holds for one root, it holds for every root. Thus, whenever $y$ is a root of $\Phi$ , $\theta(y)$ is again a root. The rational expression $\theta$ is no longer merely a relation between two selected roots; it acts on the entire finite set of roots.

This makes iteration meaningful. Starting from a root, one may form

$\displaystyle x,\quad \theta(x),\quad \theta^2(x),\quad \theta^3(x),\quad \ldots,$

where $\theta^j$ denotes repeated composition. Since there are only finitely many roots, some iterate must return to an earlier value. Choose a root lying on one of the resulting cycles, and let $n$ be its least positive period: $\theta^n(x)=x.$ The relation $\theta^n(X)-X=0$ , after clearing denominators, holds at one root and therefore at every root. Thus $\theta^n$ fixes every root. In particular, $\theta$ acts as a permutation of the roots. Moreover, every cycle has exactly the same length $n$ . If some root had a smaller period $d<n$ , then $\theta^d(X)-X$ would vanish at one root, hence at all roots, contradicting the minimality of $n$ .

The roots therefore split into equal cycles. We may write them as

$\displaystyle \theta^j(x_u),\quad 0\le j\le n-1,\quad 1\le u\le m,$

where $\mu=mn.$ The degree $\mu$ equation has been reorganized into $m$ cycles, each of length $n$ . This is the first point at which Abel’s proof ceases to be merely about formulas. The rational transformation has uncovered a hidden internal architecture in the root set.

The roots within one cycle can be regarded as a packet. Abel’s next task is to construct a quantity which remembers the packet but forgets the order of its members. Fix the cycle

$\displaystyle x_u, \theta(x_u), \theta^2(x_u),\ldots,\theta^{n-1}(x_u).$

Choose a rational expression symmetric in these $n$ members, and call its value $y_u$ . The expression may be chosen generically so that the $m$ values $y_1,\ldots,y_m$ are distinct. Concretely, one can take a sufficiently general linear combination of the elementary symmetric functions of the roots in the cycle. The point is not the particular choice; the point is that shifting once around the cycle does not change $y_u$ . Now form the power sums

$\displaystyle R_\nu=y_1^\nu+y_2^\nu+\cdots+y_m^\nu.$

Each cycle contributes the same value $y_u^\nu$ exactly $n$ times when we sum over all roots. Therefore

$\displaystyle R_\nu= \frac1n\sum_{u=1}^{m}\sum_{j=0}^{n-1}y_u^\nu.$

The right-hand side is a symmetric rational function of all $\mu$ roots. It is therefore rationally expressible in the coefficients of $\Phi$ and the known coefficients occurring in $\theta$ . Thus the quantities $R_1,\ldots,R_m$ are known. Newton’s identities then determine the elementary symmetric functions of $y_1,\ldots,y_m$ . Hence Abel obtains an equation

$\displaystyle \Psi(Y)=\prod_{u=1}^{m}(Y-y_u)=0$

of degree $m$ , with known coefficients. This is only half of the reduction. Solving the smaller equation $\Psi(Y)=0$ identifies a cycle, but one must still recover the individual roots within that cycle. For each $u$ , form the cycle polynomial

$\displaystyle P_u(X)=\prod_{j=0}^{n-1}\bigl(X-\theta^j(x_u)\bigr).$

Its coefficients are symmetric in the roots of the $u$ th cycle. Abel proves that each such coefficient is a rational function of $y_u$ . Here is the mechanism. Let $\psi(x_u)$ be one coefficient of $P_u$ . Because it is symmetric in the cycle, it has the same value if $x_u$ is replaced by any of the other roots in that cycle. Form the mixed sums

$\displaystyle T_\nu=\sum_{u=1}^{m}y_u^\nu\psi(x_u),\quad 0\le\nu\le m-1.$

Exactly as before, each $T_\nu$ is a symmetric rational function of all roots, hence known. We now seek a polynomial

$\displaystyle C(Y)=c_0+c_1Y+\cdots+c_{m-1}Y^{m-1}$

such that $\psi(x_u)=C(y_u)$ for every $u$ . The unknown coefficients $c_0,\ldots,c_{m-1}$ must satisfy

$\displaystyle T_\nu=c_0R_\nu+c_1R_{\nu+1}+\cdots+c_{m-1}R_{\nu+m-1}.$

The coefficient matrix of this system has determinant equal to the square of the Vandermonde product

$\displaystyle \prod_{1\le u<v\le m}(y_v-y_u).$

Since the $y_u$ were chosen distinct, this determinant is nonzero. The coefficients of $C$ are therefore known, and $\psi(x_u)=C(y_u)$ is indeed rational in $y_u$ . Thus Abel has achieved a precise decomposition. The original degree- $\mu=mn$ equation is reduced to a degree- $m$ equation for the cycle labels $y_u$ . Once one of these labels is known, the corresponding roots are the roots of a degree- $n$ equation $P_u(X)=0$ , whose coefficients are rational functions of that label. A single rational root transformation has therefore split one problem of degree $mn$ into a smaller outer problem of degree $m$ and cyclic inner problems of degree $n$ .

The decisive special case occurs when there is only one cycle: $m=1,\quad n=\mu.$ Then every root is obtained from one root $x$ by repeated application of a single transformation: $x, \theta(x),\theta^2(x),\ldots,\theta^{\mu-1}(x),$ with $\theta^\mu(x)=x.$ Abel now gives an explicit radical solution. This is the part of the memoir in which the structure of a cyclic permutation is converted into a radical expression. Let $\omega$ be a primitive $\mu$ th root of unity. Define the resolvents

$\displaystyle S_r=\sum_{j=0}^{\mu-1}\omega^{-rj}\theta^j(x), \quad 0\le r\le\mu-1.$

These are not arbitrary sums. They are chosen so that the cyclic transformation $\theta$ acts diagonally on them. Indeed, shifting every term once around the cycle gives $\theta(S_r)=\omega^r S_r.$ Thus, although $\theta$ permutes the roots in a complicated-looking cycle, it merely multiplies the resolvent $S_r$ by the scalar $\omega^r$ . Consequently, $S_r^\mu$ is fixed by $\theta$ . In the one-cycle case, a rational expression fixed by $\theta$ has the same value on every root. Its value is therefore symmetric in the roots and hence rationally expressible in the known coefficients. Thus $V_r:=S_r^\mu$ is known for every $r$ . At first this seems to produce many radicals: perhaps one must take a separate $\mu$ th root for every $V_r$ . Abel’s next observation is that the resolvents are not independent. After a harmless generic choice ensuring $S_1\ne0$ , define $A_r=S_rS_1^{\mu-r}.$ Applying $\theta$ gives

$\displaystyle \theta(A_r)=(\omega^rS_r)(\omega S_1)^{\mu-r}= \omega^\mu A_r=A_r.$

Thus $A_r$ is known. Since $S_1^\mu=V_1$ , we obtain

$\displaystyle S_r=\frac{A_r}{V_1}S_1^r.$

All the resolvents are therefore rational expressions in the one radical $S_1=\sqrt[\mu]{V_1}.$ The roots themselves are recovered by Fourier inversion:

$\displaystyle \theta^j(x)= \frac1\mu\sum_{r=0}^{\mu-1}\omega^{rj}S_r.$

Substituting the formulas for the $S_r$ shows that every root is a rational expression in the known coefficients, roots of unity, and one $\mu$ th root. The different roots arise by replacing $S_1$ with its different branch-values

$\displaystyle S_1, \omega S_1, \omega^2S_1,\ldots,\omega^{\mu-1}S_1.$

This is Abel’s cyclic theorem: when the roots form one rational cycle, the equation is solvable by radicals.

A useful consequence appears when $\mu$ is prime. Suppose two distinct roots are rationally related by $x'=\theta(x)$ . The common cycle length divides $\mu$ . It is not $1$ , since the transformation sends one root to a distinct one. Since $\mu$ is prime, the cycle length must be $\mu$ itself. Thus a single nontrivial rational relation between two roots of an irreducible prime-degree equation already forces the cyclic situation and hence radical solvability.

The cyclic theorem is powerful, but Abel’s real achievement is to remove the requirement that one transformation must visit every root. Suppose that all roots of the irreducible equation can be obtained rationally from one chosen root $x$ : $x, \theta_1(x), \theta_2(x),\ldots,\theta_{\mu-1}(x).$ Assume, in addition, that the corresponding transformations commute under composition:

$\displaystyle \theta_i(\theta_j(x))=\theta_j(\theta_i(x)).$

This is the hypothesis that later gave the class its name: such equations are called Abelian equations.

Choose one nontrivial transformation $\theta$ . As before, it divides the roots into cycles of a common length $n$ , producing a reduced equation $\Psi(Y)=0$ of degree $m$ , where $\mu=mn$ . The new roots $y_1,\ldots,y_m$ label the cycles. The key question is whether the smaller equation retains the same favorable structure. It does. Let $\eta$ be one of the other rational transformations. Because $\eta$ commutes with $\theta$ , it carries an entire $\theta$ -cycle to another entire $\theta$ -cycle:

$\displaystyle \eta\bigl(\theta^j(x)\bigr)=\theta^j\bigl(\eta(x)\bigr).$

Thus, if $y$ is the symmetric label attached to one cycle, then applying $\eta$ produces the label of another cycle. By the interpolation argument already used above, this new label is a rational function of $y$ . Hence $\eta$ induces a rational transformation

$\displaystyle y\longmapsto\lambda_\eta(y)$

among the roots of the smaller equation $\Psi(Y)=0$ . More importantly, the induced transformations still commute. Indeed,

$\displaystyle \lambda_\eta(\lambda_\rho(y))= \lambda_\rho(\lambda_\eta(y))$

because the original transformations $\eta$ and $\rho$ commute before the cycle reduction. Thus the degree- $m$ equation has exactly the same structural property as the original degree- $\mu$ equation: all its roots are rational functions of one root, and the corresponding transformations commute. Since $\ m<\mu,$ the procedure can be repeated. At each stage, one chooses a nontrivial transformation, separates the roots into cycles, solves a lower-degree resolvent equation, and then solves the cyclic equations attached to its roots. The degree strictly decreases at every outer reduction. Eventually it reaches degree one. Reversing the chain of reductions expresses the roots of the original equation by radicals.

Abel’s principal positive theorem is therefore this: An irreducible equation is solvable by radicals whenever all its roots are rational functions of one root and the corresponding rational transformations commute under composition.

The theorem is a sufficient criterion, not a complete characterization of all equations solvable by radicals. A single rational relation between two roots gives only one reduction; it does not guarantee that the smaller resolvent equation will itself admit compatible rational transformations. The commutativity condition is what makes the process recursive. It ensures that the structural property survives the first descent and hence survives every later descent.

The simplest natural source of commuting rational transformations comes from the multiple-angle formulas. Put $x=\cos t.$ For every positive integer $a$ , there is a polynomial $T_a$ such that $\cos(at)=T_a(\cos t).$ These polynomials satisfy

$\displaystyle T_a(T_b(x))=T_{ab}(x)=T_b(T_a(x)).$

Thus the transformations $x\mapsto T_a(x)$ commute. For an odd prime $p$ , consider the numbers $\cos\frac{2\pi}{p},\ \cos\frac{4\pi}{p},\ldots, \cos\frac{(p-1)\pi}{p}.$ They are the distinct real roots of the real cyclotomic equation of degree $\frac{p-1}{2}.$ Choose a primitive residue $g$ modulo $p$ . Multiplication by $g$ permutes the nonzero residue classes modulo $p$ ; after identifying $a$ with $-a$ , because the cosine has the same value at these two arguments, this permutation becomes one cycle on the displayed cosine values. The rational transformation responsible for it is $x\longmapsto T_g(x).$ Thus the cyclotomic equation belongs to Abel’s cyclic class. The usual radical solution of a cyclic equation is therefore not a disconnected miracle. It is the same root-of-unity and Fourier-resolvent mechanism that Abel developed abstractly. This circular example also explains the direction in which Abel wanted to push the theory. The division of the circle is governed by commuting multiplication formulas. Abel’s work on elliptic functions had revealed analogous addition and multiplication laws in a much richer setting. He saw that the same algebraic pattern—values related by rational, commuting transformations—should govern division problems for elliptic functions as well. In this sense, the memoir is not merely about a special class of polynomial equations. It is part of Abel’s larger attempt to understand how algebraic equations arise from the multiplication and division laws of transcendental functions.

We might ask if Abel’s criterion about commutativity is necessary for radical solvability. It is not. There are equations solvable by radicals whose underlying symmetries are noncommutative; the general quartic is the standard example. Abel’s theorem isolates the Abelian portion of the solvable world: equations whose root transformations themselves commute. In this class, the recursive reduction is transparent and explicit. The theorem is therefore important not because it gives the final general criterion for solvability by radicals. That criterion was supplied later by Galois. Its importance is that Abel discovered the decisive direction of the answer. Solvability does not depend primarily on the degree of an equation. It depends on the internal manner in which the roots are related and permuted. Degree is only a crude first measure. The real issue is whether the transformations connecting the roots can be dismantled into simple cyclic steps.

Modern algebra compresses Abel’s argument into a few structural statements. Let $K$ be the field generated by the known coefficients, and let $L$ be the field generated by all roots. Abel’s rational root transformations become, in modern language, symmetries of $L$ fixing $K$ . In the situation he studies, these symmetries act transitively on the roots and commute with one another. Thus they form a finite Abelian permutation group. The cyclic case is the case of a cyclic group. Abel’s resolvents $S_r=\sum_{j=0}^{\mu-1}\omega^{-rj}\theta^j(x)$ diagonalize that cyclic action: applying $\theta$ multiplies $S_r$ by $\omega^r$ . The quantities $S_r^\mu$ are invariant, while Fourier inversion reconstructs the roots. In modern language, Abel has decomposed the root space into the one-dimensional character components of a cyclic group. The general commuting case corresponds to an Abelian symmetry group. Every finite Abelian group can be decomposed into cyclic pieces, and Abel’s repeated cycle reduction is a concrete nineteenth-century version of this fact. The later general theorem says that an equation is solvable by radicals precisely when its symmetry group is solvable, meaning that it can be dismantled through successive Abelian quotients. Abel’s 1829 memoir proves the cleanest and most transparent part of this picture: Abelian symmetry itself is sufficient to force radical solvability.

The contrast with the generic quintic is now clear. The general quintic has a much larger and more intricate root symmetry. Abel’s earlier impossibility proof shows that this symmetry cannot be resolved by any finite sequence of radical branch choices. The later memoir shows the opposite phenomenon: when the symmetries among roots are commutative and reproduce themselves under reduction, radical solution becomes possible. The two works therefore belong together. One explains why a general formula fails; the other explains what kind of hidden order allows a formula to exist.

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