Abel–Ruffini theorem
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The Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general solution in radicals to polynomial equations of degree five or higher. The content of this theorem is frequently misunderstood. It does not assert that higher-degree polynomial equations are unsolvable. In fact, if the polynomial has real or complex coefficients, and we allow complex solutions, then every polynomial equation has solutions; this is the fundamental theorem of algebra. Although these solutions cannot always be computed exactly with radicals, they 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 concerns the form that such a solution must take. The content of the theorem is that the solution of a higher-degree equation cannot in all cases be expressed in terms of the polynomial coefficients with a finite number of operations of addition, subtraction, multiplication, division and root extraction. Some polynomials of arbitrary degree, of which the simplest nontrivial example is the monomial equation Failed to parse (Missing texvc executable; please see math/README to configure.): ax^n = b , are always solvable with a radical. For example, 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \textstyle{ax^2 + bx + c = 0} are
The Abel–Ruffini theorem says that there are some fifth-degree equations whose solution cannot be so expressed. The equation Failed to parse (Missing texvc executable; please see math/README to configure.): \textstyle{x^5 - x + 1 = 0} is an example. (See Bring radical.) Some other fifth degree equations can be solved by radicals, for example Failed to parse (Missing texvc executable; please see math/README to configure.): \textstyle{x^5 - x^4 - x + 1 = 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 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \textstyle{S_2, S_3}
and Failed to parse (Missing texvc executable; please see math/README to configure.): \textstyle{S_4}
are solvable groups, whereas Failed to parse (Missing texvc executable; please see math/README to configure.): \textstyle{S_n}
is not solvable for Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{n \ge 5}
.
ProofThe following proof is based on Galois theory. One of the fundamental theorems of Galois theory states that an equation is solvable in radicals if and only if it has a solvable Galois group, so the proof of the Abel-Ruffini theorem comes down to computing the Galois group of the general polynomial of the fifth degree. Let Failed to parse (Missing texvc executable; please see math/README to configure.): y_1 be a real number transcendental over the field of rational numbers Failed to parse (Missing texvc executable; please see math/README to configure.): Q , and let Failed to parse (Missing texvc executable; please see math/README to configure.): y_2 be a real number transcendental over Failed to parse (Missing texvc executable; please see math/README to configure.): Q(y_1) , and so on to Failed to parse (Missing texvc executable; please see math/README to configure.): y_5 which is transcendental over Failed to parse (Missing texvc executable; please see math/README to configure.): Q(y_1, y_2, y_3, y_4) . These numbers are called independent transcendental elements over Q. Let Failed to parse (Missing texvc executable; please see math/README to configure.): E = Q(y_1, y_2, y_3, y_4, y_5) and let
out yields the elementary symmetric functions of the Failed to parse (Missing texvc executable; please see math/README to configure.): y_n
and so on up to
The coefficient of Failed to parse (Missing texvc executable; please see math/README to configure.): x^n
in Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)
is thus Failed to parse (Missing texvc executable; please see math/README to configure.): s_{5-n}
. Because our independent transcendentals Failed to parse (Missing texvc executable; please see math/README to configure.): y_n act as indeterminates over Failed to parse (Missing texvc executable; please see math/README to configure.): Q , every permutation Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma in the symmetric group on 5 letters Failed to parse (Missing texvc executable; please see math/README to configure.): S_5 induces an automorphism Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma' on Failed to parse (Missing texvc executable; please see math/README to configure.): E that leaves Failed to parse (Missing texvc executable; please see math/README to configure.): Q fixed and permutes the elements Failed to parse (Missing texvc executable; please see math/README to configure.): y_n . Since an arbitrary rearrangement of the roots of the product form still produces the same polynomial, e.g.:
also leave Failed to parse (Missing texvc executable; please see math/README to configure.): E fixed, so they are elements of the Galois group Failed to parse (Missing texvc executable; please see math/README to configure.): G(E/F) . Now, since Failed to parse (Missing texvc executable; please see math/README to configure.): |S_5| = 5! it must be that Failed to parse (Missing texvc executable; please see math/README to configure.): |G(E/F)| \ge 5! , as there could possibly be automorphisms there that are not in Failed to parse (Missing texvc executable; please see math/README to configure.): S_5 . However, since the splitting field of a quintic polynomial has at most 5! elements, Failed to parse (Missing texvc executable; please see math/README to configure.): |G(E/F)| = 5! , and so Failed to parse (Missing texvc executable; please see math/README to configure.): G(E/F) must be isomorphic to Failed to parse (Missing texvc executable; please see math/README to configure.): S_5 . Generalizing this argument shows that the Galois group of every general polynomial of degree Failed to parse (Missing texvc executable; please see math/README to configure.): n is isomorphic to Failed to parse (Missing texvc executable; please see math/README to configure.): S_n . And what of Failed to parse (Missing texvc executable; please see math/README to configure.): S_5 ? The only composition series of Failed to parse (Missing texvc executable; please see math/README to configure.): S_5
is Failed to parse (Missing texvc executable; please see math/README to configure.): S_5 \ge A_5 \ge \{e\}
(where Failed to parse (Missing texvc executable; please see math/README to configure.): A_5
is the alternating group on five letters, also known as the icosahedral group). However, the quotient group Failed to parse (Missing texvc executable; please see math/README to configure.): A_5/\{e\}
(isomorphic to Failed to parse (Missing texvc executable; please see math/README to configure.): A_5
itself) is not an abelian group, and so Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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. Failed to parse (Missing texvc executable; please see math/README to configure.): 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. 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. HistoryAround 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. 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 and contained a minor gap. Still it was quite innovative in using permutation groups. The theorem is generally credited to Niels Henrik Abel, who published a proof in 1824. Insights into these issues were also gained using Galois theory pioneered by Évariste Galois. In 1885, John Stuart Glashan, George Paxton Young, and Carl Runge provided a proof using this theory. See alsoReferences
es:Teorema de Abel-Ruffini fr:Théorème d'Abel (algèbre) it:Teorema di Abel-Ruffini nl:Stelling van Abel-Ruffini pl:Twierdzenie Abela-Ruffiniego pt:Teorema de Abel-Ruffini ru:Теорема Абеля — Руффини fi:Abelin-Ruffinin lause tr:Abel Teoremi |
