Abel–Ruffini theorem
Encyclopedia
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.
. Although the solutions cannot always be expressed 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 theorem says that not all solutions of higher-degree equations can be obtained by starting with the equation's coefficients and rational
constants, and repeatedly forming sums, differences, products, quotients, and radicals (n-th roots, for some integer n) of previously obtained numbers. This clearly excludes the possibility of having any formula that expresses the solutions of an arbitrary equation of degree 5 or higher in terms of its coefficients, using only those operations, or even of having different formulas for different roots or for different classes of polynomials, in such a way as to cover all cases. (In principle one could imagine formulas using irrational numbers as constants, but even if a finite number of those were admitted at the start, not all roots of higher-degree equations could be obtained.) However some polynomial equations, of arbitrarily high degree, are solvable with such operations. Indeed if the roots happen to be rational numbers, they can trivially be expressed as constants. The simplest nontrivial example is the equation
, whose solutions are
Here the expression
, which appears to involve the use of the exponential function, in fact just gives the different possible values of 
(the n-th roots of unity), so it involves only extraction of radicals.
s, using the familiar quadratic formula: The roots of the following equation are shown below:
Analogous formulas for third- and fourth-degree equations, using cube roots and fourth roots, had been known since the 16th century.
equations whose solution cannot be so expressed. The equation
is an example. (See Bring radical
.) Some other fifth degree equations can be solved by radicals, for example
, which factorizes to 
. 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 group
s 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.
. Historically, Ruffini and Abel's proofs precede 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
be a real number
transcendental
over the field of rational number
s
, and let 
be a real number transcendental over 
, and so on to 
which is transcendental over 
. These numbers are called independent transcendental elements over Q. Let 
and let
Multiplying
out yields the elementary symmetric function
s of the
:
The coefficient of
in 
is thus 
. Because our independent transcendentals 
act as indeterminates over 
, every permutation 
in the symmetric group on 5 letters 
induces an automorphism
on 
that leaves 
fixed and permutes the elements 
. Since an arbitrary rearrangement of the roots of the product form still produces the same polynomial, e.g.:
is still the same polynomial as
the automorphisms
also leave 
fixed, so they are elements of the Galois group 
. Now, since 
it must be that 
, as there could possibly be automorphisms there that are not in 
.
However, since the splitting field
of a quintic polynomial has at most 5! elements,
, and so 
must be isomorphic
to
. Generalizing this argument shows that the Galois group of every general polynomial of degree 
is isomorphic to 
.
And what of
? The only composition series
of
is 
(where 
is the alternating group on five letters, also known as the icosahedral group). However, the quotient group
(isomorphic to 
itself) is not an abelian group
, and so
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
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.
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.
Q.E.D.
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 permutation
s, 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. Ruffini assumed that a solution would necessarily be a function of the radicals (in modern terms, he failed to prove that the splitting field is one of the fields in the tower of radicals which corresponds to a solution expressed in radicals). 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.
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.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general algebraic solution
Algebraic solution
An algebraic solution is a closed form expression that is the solution of an algebraic equation in terms of the coefficients, relying only on addition, subtraction, multiplication, division, and the extraction of roots ....
—that is, solution in radicals
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...
— to polynomial equations of degree five or higher.
Interpretation
The content of this theorem is frequently misunderstood. It does not assert that higher-degree polynomial equations are unsolvable. 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 solution; this is the fundamental theorem of algebraFundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
. Although the solutions cannot always be expressed exactly with radicals, they can be computed to any desired degree of accuracy using numerical methods such as the Newton–Raphson method
Newton's method
In numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's 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 theorem says that not all solutions of higher-degree equations can be obtained by starting with the equation's coefficients and rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
constants, and repeatedly forming sums, differences, products, quotients, and radicals (n-th roots, for some integer n) of previously obtained numbers. This clearly excludes the possibility of having any formula that expresses the solutions of an arbitrary equation of degree 5 or higher in terms of its coefficients, using only those operations, or even of having different formulas for different roots or for different classes of polynomials, in such a way as to cover all cases. (In principle one could imagine formulas using irrational numbers as constants, but even if a finite number of those were admitted at the start, not all roots of higher-degree equations could be obtained.) However some polynomial equations, of arbitrarily high degree, are solvable with such operations. Indeed if the roots happen to be rational numbers, they can trivially be expressed as constants. The simplest nontrivial example is the equation
, whose solutions areHere the expression
, which appears to involve the use of the exponential function, in fact just gives the different possible values of (the n-th roots of unity), so it involves only extraction of radicals.Lower-degree polynomials
The solutions of any second-degree polynomial equation can be expressed in terms of addition, subtraction, multiplication, division, and square rootSquare root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
s, using the familiar quadratic formula: The roots of the following equation are shown below:
Analogous formulas for third- and fourth-degree equations, using cube roots and fourth roots, had been known since the 16th century.
Quintics and higher
The Abel–Ruffini theorem says that there are some fifth-degreeQuintic equation
In mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...
equations whose solution cannot be so expressed. The equation
is an example. (See Bring radicalBring radical
In algebra, a Bring radical or ultraradical of a complex number a is a root of the polynomialx^5+x+a. \,In algebra, a Bring radical or ultraradical of a complex number a is a root of the polynomialx^5+x+a. \,In algebra, a Bring radical or ultraradical of a complex number a is a root...
.) Some other fifth degree equations can be solved by radicals, for example
, which factorizes to . The precise criterion that distinguishes between those equations that can be solved by radicals and those that cannot was given by Évariste GaloisÉvariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
and is now part of Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
: a polynomial equation can be solved by radicals if and only if its Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
(over the rational numbers, or more generally over the base field of admitted constants) is a solvable group
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
.
Today, in the modern algebraic
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
context, we say that second, third and fourth degree polynomial equations can always be solved by radicals because the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
s 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
Indeterminate (variable)
In mathematics, and particularly in formal algebra, an indeterminate is a symbol that does not stand for anything else but itself. In particular it does not designate a constant, or a parameter of the problem, it is not an unknown that could be solved for, it is not a variable designating a...
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
Algebraic independence
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K...
values over the base field.
Proof
The following proof is based on Galois theoryGalois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
. Historically, Ruffini and Abel's proofs precede Galois theory.
One of the fundamental theorems of Galois theory states that an equation is solvable in radicals if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it has a solvable Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
, so the proof of the Abel–Ruffini theorem comes down to computing the Galois group of the general polynomial of the fifth degree.
Let
be a real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
over the field of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s
, and let be a real number transcendental over , and so on to which is transcendental over . These numbers are called independent transcendental elements over Q. Let and letMultiplying
out yields the elementary symmetric functionSymmetric function
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...
s of the
:The coefficient of
in is thus . Because our independent transcendentals act as indeterminates over , every permutation in the symmetric group on 5 letters induces an automorphismAutomorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
on that leaves fixed and permutes the elements . Since an arbitrary rearrangement of the roots of the product form still produces the same polynomial, e.g.:is still the same polynomial as
the automorphisms
also leave fixed, so they are elements of the Galois group . Now, since it must be that , as there could possibly be automorphisms there that are not in .However, since the splitting field
Splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...
of a quintic polynomial has at most 5! elements,
, and so must be isomorphicIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to
. Generalizing this argument shows that the Galois group of every general polynomial of degree is isomorphic to .And what of
? The only composition seriesComposition series
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence...
of
is (where is the alternating group on five letters, also known as the icosahedral group). However, the quotient groupQuotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
(isomorphic to itself) is not an abelian groupAbelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
, and so
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 subgroupNormal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of the symmetric group on n letters is always the alternating group on n letters, and since the alternating groups on n letters for
are always simpleSimple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
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.
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.Q.E.D.
History
Around 1770, Joseph Louis LagrangeJoseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
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 permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
s, 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
Paolo Ruffini
Paolo Ruffini was an Italian mathematician and philosopher.By 1788 he had earned university degrees in philosophy, medicine/surgery, and mathematics...
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. Ruffini assumed that a solution would necessarily be a function of the radicals (in modern terms, he failed to prove that the splitting field is one of the fields in the tower of radicals which corresponds to a solution expressed in radicals). 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
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
, who published a proof that required just six pages 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.