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In algebra, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher with arbitrary coefficients.[1] The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799, and Niels Henrik Abel, who provided a proof in 1823. Évariste Galois independently proved the theorem in a work that was posthumously published in 1846.[2]

Interpretation

The theorem does not assert that some higher-degree polynomial equations have no solution. In fact, the opposite is true: every non-constant polynomial equation in one unknown, with real or complex coefficients, has at least one complex number as a solution (and thus, by polynomial division, as many complex roots as its degree, counting repeated roots); this is the fundamental theorem of algebra. These solutions can be computed to any desired degree of accuracy using numerical methods such as the Newton–Raphson method or Laguerre method, and in this way they are no different from solutions to polynomial equations of the second, third, or fourth degrees. The theorem only shows that the solutions of some of these equations cannot be expressed via a general expression in radicals.

Also, the theorem does not assert that some higher-degree polynomial equations have roots which cannot be expressed in terms of radicals. While this is now known to be true, it is a stronger claim, which was only proven a few years later by Galois. The theorem only shows that there is no general solution in terms of radicals which gives the roots to a generic polynomial with arbitrary coefficients. It did not by itself rule out the possibility that each polynomial may be solved in terms of radicals on a case-by-case basis.
Lower-degree polynomials

The solutions of any second-degree polynomial equation can be expressed in terms of addition, subtraction, multiplication, division, and square roots, using the familiar quadratic formula: The roots of the following equation are shown below:

\( \textstyle{ax^2 + bx + c = 0, a \ne 0} \)

\( x=\frac{-b \pm \sqrt {b^2-4ac\ }}{2a}. \)

Analogous formulas for third- and fourth-degree equations, using cube roots and fourth roots, have been known since the 16th century.

Quintics and higher

The Abel–Ruffini theorem says that there are some fifth-degree equations whose solution cannot be so expressed. The equation \( x^5 - x + 1 = 0 \) is an example. (See Bring radical.) Some other fifth degree equations can be solved by radicals, for example \( x^5- x^4 - x + 1 = 0 \), which factors into \( (x-1)(x-1)(x+1)(x+i)(x-i) = 0 \) . The precise criterion that distinguishes between those equations that can be solved by radicals and those that cannot was given by Évariste Galois and is now part of Galois theory: a polynomial equation can be solved by radicals if and only if its Galois group (over the rational numbers, or more generally over the base field of admitted constants) is a solvable group.

Today, in the modern algebraic context, we say that second, third and fourth degree polynomial equations can always be solved by radicals because the symmetric groups S2, S3 and S4 are solvable groups, whereas Sn is not solvable for n ≥ 5. This is so because for a polynomial of degree n with indeterminate coefficients (i.e., given by symbolic parameters), the Galois group is the full symmetric group Sn (this is what is called the "general equation of the n-th degree"). This remains true if the coefficients are concrete but algebraically independent values over the base field.

Proof

The following proof is based on Galois theory (for a short explanation of Arnold's proof that does not rely on prior knowledge in group theory see [3]). Historically, Ruffini and Abel's proofs precede Galois theory and Arnold's. For a modern presentation of Abel's proof see the books of Tignol or Pesic.

One of the fundamental theorems of Galois theory states that a polynomial f(x) ∈ F[x] is solvable by radicals over F if and only if its splitting field K over F has a solvable Galois group,[4] so the proof of the Abel–Ruffini theorem comes down to computing the Galois group of the general polynomial of the fifth degree.

Let \( y_1 \) be a real number transcendental over the field of rational numbers Q, and let \( y_2 \) be a real number transcendental over \( Q(y_1) \) , and so on to \( y_5 \) which is transcendental over \( Q(y_1, y_2, y_3, y_4) \) . These numbers are called independent transcendental elements over Q. Let \( E = Q(y_1, y_2, y_3, y_4, y_5) \) and let

\( f(x) = (x - y_1)(x - y_2)(x - y_3)(x - y_4)(x - y_5) \in E[x]. \)

Expanding f(x) out yields the elementary symmetric functions of the \( y_n: \)

\( s_1 = y_1 + y_2 + y_3 + y_4 + y_5 \)
\( s_2 = y_1y_2 + y_1y_3 + y_1y_4 + y_1y_5 + y_2y_3 + y_2y_4 + y_2y_5 + y_3y_4 + y_3y_5 + y_4y_5 \)
\( s_3 = y_1y_2y_3 + y_1y_2y_4 + y_1y_2y_5 + y_1y_3y_4 + y_1y_3y_5 + y_1y_4y_5 +y_2y_3y_4 + y_2y_3y_5 + y_2y_4y_5 + y_3y_4y_5 \)
\( s_4 = y_1y_2y_3y_4 + y_1y_2y_3y_5 + y_1y_2y_4y_5 + y_1y_3y_4y_5 + y_2y_3y_4y_5 \)
\( s_5 = y_1y_2y_3y_4y_5. \)

The coefficient of \( x^n \) in f(x) is thus \( (-1)^{5-n} s_{5-n} \) . Let \( F = Q(s_i) \) be the field obtained by adjoining the symmetric functions to the rationals (the \( s_i are all transcendental, because the \( y_i \) are independent). Because our independent transcendentals \( y_n \) act as indeterminates over Q, every permutation \( \sigma i \) n the symmetric group on 5 letters \( S_5 \) induces a distinct automorphism \( \sigma' \) on E that leaves Q fixed and permutes the elements \( y_n \) . Since an arbitrary rearrangement of the roots of the product form still produces the same polynomial, e.g.:

\( (y - y_3)(y - y_1)(y - y_2)(y - y_5)(y - y_4) \)

is still the same polynomial as

\( (y - y_1)(y - y_2)(y - y_3)(y - y_4)(y - y_5) \)

the automorphisms \( \sigma' \) also leave f fixed, so they are elements of the Galois group G(E/F). So we have shown that \( S_5 \subseteq G(E/F) \) ; however there could possibly be automorphisms there that are not in \( S_5 \) . However, since the relative automorphism group for the splitting field of a quintic polynomial has at most 5! elements, it follows that G(E/F) is isomorphic to \( S_5 \). Generalizing this argument shows that the Galois group of every general polynomial of degree n is isomorphic to \( S_n. \)

And what of \( S_5 \) ? The only composition series of \( S_5 \) is \( S_5 \ge A_5 \ge \{e\} \) (where \( A_5 \) is the alternating group on five letters, also known as the icosahedral group). However, the quotient group \( A_5/\{e\} \) (isomorphic to \( A_5 \) itself) is not an abelian group, and so \( S_5 \) 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 n letters is always the alternating group on n letters, and since the alternating groups on n letters for \( n \ge 5 \) 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.

Note that 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, e.g. \( x^5 - 1 \) 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 \( S_5 \) or \( A_5 \) 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. Q.E.D.

History

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. 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 theorem, however, was first nearly proved by Paolo Ruffini in 1799, but his proof was mostly ignored. He had several times tried to send it to different mathematicians to get it acknowledged, amongst them, French mathematician Augustin-Louis Cauchy, but it was never acknowledged, possibly because the proof was spanning 500 pages. The proof also, as was discovered later, contained an error. In modern terms, Ruffini failed to prove that the splitting field is one of the fields in the tower of radicals which corresponds to the hypothesized solution by radicals; this assumption fails, for example, for Cardano's solution of the cubic; it splits not only the original cubic but also the two others with the same discriminant. While Cauchy felt that the assumption was minor, most historians believe that the proof was not complete until Abel proved this assumption. The theorem is thus generally credited to Niels Henrik Abel, who published a proof that required just six pages in 1824.[5]

Proving that some 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; for example x^5-1=0 is solvable. Abel was working on a complete characterization when he died in 1829.[6] Furthermore, Ian Stewart notes that "for all that Abel's methods could prove, every particular quintic equation might be soluble, with a special formula for each equation."[7]

In 1830 Galois (at the age of 18) submitted to the Paris Academy of Sciences a memoir on his theory of solvability by radicals; Galois' paper 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 then died in 1832 and his paper—"Memoire sur les conditions de resolubilite des equations par radicaux"—remained unpublished until 1846 when it was published by Joseph Liouville accompanied by some of his own explanations.[6] Prior to this publication, Liouville announced Galois' result to the Academy in a speech he gave on 4 July 1843.[8] According to Allan Clark, Galois's characterization "dramatically supersedes the work of Abel and Ruffini."[9]

In 1963, Vladimir Arnold discovered a topological proof of the Abel–Ruffini theorem,[10] which served as a starting point for topological Galois theory.[11]