Abel–Ruffini theorem

Polynomial equations of degree two can be solved with the quadratic formula, which has been known since antiquity. Similarly the cubic formula for degree three, and the quartic formula for degree four, were found during the 16th century. At that time a fundamental problem was whether equations of higher degree could be solved in a similar way.

The fact that every polynomial equation of positive degree has solutions, possibly non-real, was asserted during the 17th century, but completely proved only at the beginning of the 19th century. This is the fundamental theorem of algebra. This theorem does not provide any tool for computing exactly the solutions, but Newton's method allows approximating them to any desired accuracy.

From the 16th century to beginning of the 19th century, the main problem of algebra was to search for a formula for the solutions of polynomial equations of degree five and higher, hence the name the "fundamental theorem of algebra". This meant a solution in radicals, that is, an expression involving only the coefficients of the equation, and the operations of addition, subtraction, multiplication, division, and nth root extraction.

The Abel–Ruffini theorem proves that this is impossible. However, this does not imply that a specific equation of any degree cannot be solved in radicals. On the contrary, there are equations of any degree that can be solved in radicals. This is the case of the equation ${\displaystyle x^{n}-1=0}$ for any n, and the equations defined by cyclotomic polynomials, whose all solutions can be expressed in radicals.

Abel's proof of the theorem does not explicitly contain the assertion that there are specific equations that cannot be solved by radicals. Such an assertion is not a consequence of Abel's statement of the theorem, as the statement does not exclude the possibility that "every particular quintic equation might be soluble, with a special formula for each equation."[4] However, the existence of specific equations that cannot be solved in radicals seems to be a consequence of Abel's proof, as the proof uses the fact that some polynomials in the coefficients are not the zero polynomial, and, given a finite number of polynomials, there are values of the variables at which none of the polynomials takes the value zero.

Soon after Abel's publication of its proof, Évariste Galois introduced a theory, now called Galois theory that allows deciding, for any given equation, whether it is solvable in radicals (this is theoretical, as, in practice, this decision may need huge computation which can be difficult, even with powerful computers). This decision is done by introducing auxiliary polynomials, called resolvents, whose coefficients depend polynomially upon those of the original polynomial. The polynomial is solvable in radicals if and only if some resolvent has a rational root.

The following proof is based on Galois theory and it is valid for any field of characteristic 0. Historically, Ruffini[1] and Abel's proofs precede Galois theory. For a modern presentation of Abel's proof see the article of Rosen[5] or the books of Tignol[6] or Pesic.[7]

One of the fundamental theorems of Galois theory states that a polynomial ${\displaystyle P(x)\in F\left[x\right]}$ is solvable by radicals over ${\displaystyle F}$ if and only if its splitting field ${\displaystyle K}$ over ${\displaystyle F}$ has a solvable Galois group,[8] so the proof of the Abel–Ruffini theorem comes down to computing the Galois group of the general polynomial of the fifth degree, and showing that it is not solvable.

Consider five indeterminates ${\displaystyle y_{1},y_{2},y_{3},y_{4}}$, and ${\displaystyle y_{5}}$, let ${\displaystyle E=\mathbf {Q} \left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)}$, and let

${\displaystyle P(x)=\left(x-y_{1}\right)\left(x-y_{2}\right)\left(x-y_{3}\right)\left(x-y_{4}\right)\left(x-y_{5}\right)\in E[x]}$.

Expanding ${\displaystyle P(x)}$ out yields the elementary symmetric functions of the ${\displaystyle y_{i}}$:

${\displaystyle s_{1}=y_{1}+y_{2}+y_{3}+y_{4}+y_{5}}$,

${\displaystyle s_{2}=y_{1}y_{2}+y_{1}y_{3}+y_{1}y_{4}+y_{1}y_{5}+y_{2}y_{3}+y_{2}y_{4}+y_{2}y_{5}+y_{3}y_{4}+y_{3}y_{5}+y_{4}y_{5}}$,

${\displaystyle s_{3}=y_{1}y_{2}y_{3}+y_{1}y_{2}y_{4}+y_{1}y_{2}y_{5}+y_{1}y_{3}y_{4}+y_{1}y_{3}y_{5}+y_{1}y_{4}y_{5}+y_{2}y_{3}y_{4}+y_{2}y_{3}y_{5}+y_{2}y_{4}y_{5}+y_{3}y_{4}y_{5}}$,

${\displaystyle s_{4}=y_{1}y_{2}y_{3}y_{4}+y_{1}y_{2}y_{3}y_{5}+y_{1}y_{2}y_{4}y_{5}+y_{1}y_{3}y_{4}y_{5}+y_{2}y_{3}y_{4}y_{5}}$,

${\displaystyle s_{5}=y_{1}y_{2}y_{3}y_{4}y_{5}}$.

The coefficient of ${\displaystyle x^{n}}$ in ${\displaystyle P(x)}$ is thus ${\displaystyle (-1)^{5-n}s_{5-n}}$. Let ${\displaystyle F=\mathbf {Q} (s_{1},s_{2},s_{3},s_{4},s_{5})}$ be the field obtained by adjoining the elementary symmetric functions to the rationals. Then ${\displaystyle P(x)\in F\left[x\right]}$. Because the ${\displaystyle y_{i}}$'s are indeterminates, every permutation ${\displaystyle \sigma }$ in the symmetric group on 5 letters ${\displaystyle S_{5}}$ induces a distinct automorphism ${\displaystyle \sigma '}$ on ${\displaystyle E}$ that leaves ${\displaystyle \mathbf {Q} }$ fixed and permutes the elements ${\displaystyle y_{i}}$. Since an arbitrary rearrangement of the roots of the product form still produces the same polynomial, for example,

${\displaystyle \left(x-y_{3}\right)\left(x-y_{1}\right)\left(x-y_{2}\right)\left(x-y_{5}\right)\left(x-y_{4}\right)}$

is the same polynomial as

${\displaystyle \left(x-y_{1}\right)\left(x-y_{2}\right)\left(x-y_{3}\right)\left(x-y_{4}\right)\left(x-y_{5}\right)}$,

the automorphisms ${\displaystyle \sigma '}$ also leave ${\displaystyle F}$ fixed, so they are elements of the Galois group ${\displaystyle \operatorname {Gal} (E/F)}$. Therefore, we have shown that ${\displaystyle S_{5}\subseteq \operatorname {Gal} (E/F)}$; however there could possibly be automorphisms there that are not in ${\displaystyle S_{5}}$. But, since the Galois group of the splitting field of a quintic polynomial has at most ${\displaystyle 5!}$ elements, and since ${\displaystyle E}$ is a splitting field of ${\displaystyle P(x)}$, it follows that ${\displaystyle \operatorname {Gal} (E/F)}$ is isomorphic to ${\displaystyle S_{5}}$. Generalizing this argument shows that the Galois group of every general polynomial of degree ${\displaystyle n}$ is isomorphic to ${\displaystyle S_{n}}$.

The only composition series of ${\displaystyle S_{5}}$ is ${\displaystyle S_{5}\geq A_{5}\geq \lbrace e\rbrace }$ (where ${\displaystyle A_{5}}$ is the alternating group on five letters, also known as the icosahedral group). However, the quotient group ${\displaystyle A_{5}/\lbrace e\rbrace }$ (isomorphic to ${\displaystyle A_{5}}$ itself) is not abelian, and so ${\displaystyle S_{5}}$ is not solvable, so it must be that the general polynomial of the fifth degree has no solution in radicals. Since the first nontrivial normal subgroup of the symmetric group on ${\displaystyle n}$ letters is always the alternating group on ${\displaystyle n}$ letters, and since the alternating groups on ${\displaystyle n}$ letters for ${\displaystyle n\geq 5}$ are always simple and non-abelian, and hence not solvable, it also says that the general polynomials of all degrees higher than the fifth also have no solution in radicals. Q.E.D.

The above construction of the Galois group for a fifth degree polynomial only applies to the general polynomial; specific polynomials of the fifth degree may have different Galois groups with quite different properties, for example, ${\displaystyle x^{5}-1}$ has a splitting field generated by a primitive 5th root of unity, and hence its Galois group is abelian and the equation itself solvable by radicals; moreover, the argument does not provide any rational-valued quintic that has ${\displaystyle S_{5}}$ or ${\displaystyle A_{5}}$ as its Galois group. However, since the result is on the general polynomial, it does say that a general "quintic formula" for the roots of a quintic using only a finite combination of the arithmetic operations and radicals in terms of the coefficients is impossible.

The proof is not valid if applied to polynomials whose degree is less than 5. Indeed:

• the group ${\displaystyle A_{4}}$ is not simple, because the subgroup ${\displaystyle \{e,(12)(34),(13)(24),(14)(23)\}}$, isomorphic to the Klein four-group, is a normal subgroup;
• the groups ${\displaystyle A_{2}}$ and ${\displaystyle A_{3}}$ are simple, but since they are abelian too (${\displaystyle A_{2}}$ is the trivial group and ${\displaystyle A_{3}}$ is the cyclic group of order 3), that is not a problem.

The proof remains valid if, instead of working with five indeterminates, one works with five concrete algebraically independent complex numbers, because, by the same argument, ${\displaystyle \operatorname {Gal} (E/F)=S_{5}}$.

Around 1770, Joseph Louis Lagrange began the groundwork that unified the many different tricks that had been used up to that point to solve equations, relating them to the theory of groups of permutations, in the form of Lagrange resolvents.[9] This innovative work by Lagrange was a precursor to Galois theory, and its failure to develop solutions for equations of fifth and higher degrees hinted that such solutions might be impossible, but it did not provide conclusive proof. The first person who conjectured that the problem of solving quintics by radicals might be impossible to solve was Carl Friedrich Gauss, who wrote in 1798 in section 359 of his book Disquisitiones Arithmeticae (which would be published only in 1801) that "there is little doubt that this problem does not so much defy modern methods of analysis as that it proposes the impossible". The next year, in his thesis, he wrote "After the labors of many geometers left little hope of ever arriving at the resolution of the general equation algebraically, it appears more and more likely that this resolution is impossible and contradictory." And he added "Perhaps it will not be so difficult to prove, with all rigor, the impossibility for the fifth degree. I shall set forth my investigations of this at greater length in another place." Actually, Gauss published nothing else on this subject.[1]

Paolo Ruffini, Teoria generale delle equazioni, 1799

The theorem was first nearly proved by Paolo Ruffini in 1799.[10] He sent his proof to several mathematicians to get it acknowledged, amongst them Lagrange (who did not reply) and Augustin-Louis Cauchy, who sent him a letter saying: "Your memoir on the general solution of equations is a work which I have always believed should be kept in mind by mathematicians and which, in my opinion, proves conclusively the algebraic unsolvability of general equations of higher than fourth degree."[11] However, in general, Ruffini's proof was not considered convincing. Abel wrote: "The first and, if I am not mistaken, the only one who, before me, has sought to prove the impossibility of the algebraic solution of general equations is the mathematician Ruffini. But his memoir is so complicated that it is very difficult to determine the validity of his argument. It seems to me that his argument is not completely satisfying."[11][12]

The proof also, as it was discovered later, was incomplete. Ruffini assumed that all radicals that he was dealing with could be expressed from the roots of the polynomial using field operations alone; in modern terms, he assumed that the radicals belonged to the splitting field of the polynomial. To see why this is really an extra assumption, consider, for instance, the polynomial ${\displaystyle P(x)=x^{3}-15x-20}$. According to Cardano's formula, one of its roots (all of them, actually) can be expressed as the sum of a cube root of ${\displaystyle 10+5i}$ with a cube root of ${\displaystyle 10-5i}$. On the other hand, since ${\displaystyle P(-3)<0}$, ${\displaystyle P(-2)>0}$, ${\displaystyle P(-1)<0}$, and ${\displaystyle P(5)>0}$, the roots ${\displaystyle r_{1}}$, ${\displaystyle r_{2}}$, and ${\displaystyle r_{3}}$ of ${\displaystyle P(x)}$ are all real and therefore the field ${\displaystyle \mathbf {Q} (r_{1},r_{2},r_{3})}$ is a subfield of ${\displaystyle \mathbf {R} }$. But then the numbers ${\displaystyle 10\pm 5i}$ cannot belong to ${\displaystyle \mathbf {Q} (r_{1},r_{2},r_{3})}$. While Cauchy either did not notice Ruffini's assumption or felt that it was a minor one, most historians believe that the proof was not complete until Abel proved the theorem on natural irrationalities, which asserts that the assumption holds in the case of general polynomials.[6][13] The Abel–Ruffini theorem is thus generally credited to Abel, who published a proof compressed into just six pages in 1824.[2] (Abel adopted a very terse style to save paper and money: the proof was printed at his own expense.[7]) A more elaborated version of the proof would be published in 1826.[3]

Proving that the general quintic (and higher) equations were unsolvable by radicals did not completely settle the matter, because the Abel–Ruffini theorem does not provide necessary and sufficient conditions for saying precisely which quintic (and higher) equations are unsolvable by radicals. Abel was working on a complete characterization when he died in 1829.[14]

According to Nathan Jacobson, "The proofs of Ruffini and of Abel [...] were soon superseded by the crowning achievement of this line of research: Galois' discoveries in the theory of equations."[8] In 1830, Galois (at the age of 18) submitted to the Paris Academy of Sciences a memoir on his theory of solvability by radicals, which was ultimately rejected in 1831 as being too sketchy and for giving a condition in terms of the roots of the equation instead of its coefficients. Galois was aware of the contributions of Ruffini and Abel, since he wrote "It is a common truth, today, that the general equation of degree greater than 4 cannot be solved by radicals... this truth has become common (by hearsay) despite the fact that geometers have ignored the proofs of Abel and Ruffini..."[1] Galois then died in 1832 and his paper Mémoire sur les conditions de resolubilité des équations par radicaux[15] remained unpublished until 1846, when it was published by Joseph Liouville accompanied by some of his own explanations.[14] Prior to this publication, Liouville announced Galois' result to the Academy in a speech he gave on 4 July 1843.[4] A simplification of Abel's proof was published by Pierre Wantzel in 1845.[16] When he published it, he was already aware of the contributions by Galois and he mentions that, whereas Abel's proof is valid only for general polynomials, Galois' approach can be used to provide a concrete polynomial of degree 5 whose roots cannot be expressed in radicals from its coefficients.

In 1963, Vladimir Arnold discovered a topological proof of the Abel–Ruffini theorem,[17][18][19] which served as a starting point for topological Galois theory.[20]