Abel–Ruffini theorem

In mathematics, the Abel–Ruffini theorem states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equation are viewed and manipulated as indeterminates.
The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799 and Niels Henrik Abel, who provided a proof in 1824.
The term can also refer to the slightly stronger result that there are equations of degree five and higher that cannot be solved by radicals. This does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients of the equation are not the zero polynomial. This improved statement follows directly from. Galois theory implies also that
is the simplest equation that cannot be solved in radicals, and that almost all polynomials of degree five or higher cannot be solved in radicals.
The impossibility of solving in degree five or higher contrasts with the case of lower degree: one has the quadratic formula, the cubic formula, and the quartic formula for degrees two, three, and four, respectively.

Context

s 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, which does not provide any tool for computing the solutions, although several methods are known for approximating all solutions to any desired accuracy.
From the 16th century to the 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 th root extraction.
The Abel–Ruffini theorem proves that this is impossible. However, this impossibility 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 for any, and the equations defined by cyclotomic polynomials, all of whose 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." 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 his proof, Évariste Galois introduced a theory, now called Galois theory, that allows deciding, for any given equation, whether it is solvable in radicals. This was purely theoretical before the rise of electronic computers. With modern computers and programs, deciding whether a polynomial is solvable by radicals can be done for polynomials of degree greater than 100. Computing the solutions in radicals of solvable polynomials requires huge computations. Even for the degree five, the expression of the solutions is so huge that it has no practical interest.

Proof

The proof of the Abel–Ruffini theorem predates Galois theory. However, Galois theory allows a better understanding of the subject, and modern proofs are generally based on it, while the original proofs of the Abel–Ruffini theorem are still presented for historical purposes.
The proofs based on Galois theory comprise four main steps: the characterization of solvable equations in terms of field theory; the use of the Galois correspondence between subfields of a given field and the subgroups of its Galois group for expressing this characterization in terms of solvable groups; the proof that the symmetric group is not solvable if its degree is five or higher; and the existence of polynomials with a symmetric Galois group.

Algebraic solutions and field theory

An algebraic solution of a polynomial equation is an expression involving the four basic arithmetic operations, and root extractions. Such an expression may be viewed as the description of a computation that starts from the coefficients of the equation to be solved and proceeds by computing some numbers, one after the other.
At each step of the computation, one may consider the smallest field that contains all numbers that have been computed so far. This field is changed only for the steps involving the computation of an th root.
So, an algebraic solution produces a sequence
of fields, and elements such that
for with for some integer An algebraic solution of the initial polynomial equation exists if and only if there exists such a sequence of fields such that contains a solution.
For having normal extensions, which are fundamental for the theory, one must refine the sequence of fields as follows. If does not contain all -th roots of unity, one introduces the field that extends by a primitive root of unity, and one redefines as
So, if one starts from a solution in terms of radicals, one gets an increasing sequence of fields such that the last one contains the solution, and each is a normal extension of the preceding one with a Galois group that is cyclic.
Conversely, if one has such a sequence of fields, the equation is solvable in terms of radicals. For proving this, it suffices to prove that a normal extension with a cyclic Galois group can be built from a succession of radical extensions.

Galois correspondence

The Galois correspondence establishes a one to one correspondence between the subextensions of a normal field extension and the subgroups of the Galois group of the extension. This correspondence maps a field such to the Galois group of the automorphisms of that leave fixed, and, conversely, maps a subgroup of to the field of the elements of that are fixed by.
The preceding section shows that an equation is solvable in terms of radicals if and only if the Galois group of its splitting field is solvable, that is, it contains a sequence of subgroups such that each is normal in the preceding one, with a quotient group that is cyclic..
So, for proving the Abel–Ruffini theorem, it remains to show that the symmetric group is not solvable, and that there are polynomials with symmetric Galois groups.

Solvable symmetric groups

For, the symmetric group of degree has only the alternating group as a nontrivial normal subgroup. For, the alternating group is simple and not abelian. This implies that both and are not solvable for. Thus, the Abel–Ruffini theorem results from the existence of polynomials with a symmetric Galois group; this will be shown in the next section.
On the other hand, for, the symmetric group and all its subgroups are solvable. This explains the existence of the quadratic, cubic, and quartic formulas, since a major result of Galois theory is that a polynomial equation has a solution in radicals if and only if its Galois group is solvable.

Polynomials with symmetric Galois groups

General equation

The general or generic polynomial equation of degree is the equation
where are distinct indeterminates. This is an equation defined over the field of the rational fractions in with rational number coefficients. The original Abel–Ruffini theorem asserts that, for, this equation is not solvable in radicals. In view of the preceding sections, this results from the fact that the Galois group over of the equation is the symmetric group .
For proving that the Galois group is it is simpler to start from the roots. Let be new indeterminates, aimed to be the roots, and consider the polynomial
Let be the field of the rational fractions in and be its subfield generated by the coefficients of The permutations of the induce automorphisms of. Vieta's formulas imply that every element of is a symmetric function of the and is thus fixed by all these automorphisms. It follows that the Galois group is the symmetric group
The fundamental theorem of symmetric polynomials implies that the are algebraic independent, and thus that the map that sends each to the corresponding is a field isomorphism from to. This means that one may consider as a generic equation. This finishes the proof that the Galois group of a general equation is the symmetric group, and thus proves the original Abel–Ruffini theorem, which asserts that the general polynomial equation of degree cannot be solved in radicals for.

Explicit example

The equation is not solvable in radicals, as will be explained below.
Let be.
Let be its Galois group, which acts faithfully on the set of complex roots of.
Numbering the roots lets one identify with a subgroup of the symmetric group.
Since factors as in, the group contains a permutation that is a product of disjoint cycles of lengths 2 and 3 ; then also contains, which is a transposition. Since is irreducible in, the same principle shows that contains a 5-cycle. Because 5 is prime, any transposition and 5-cycle in generate the whole group; see. Thus. Since the group is not solvable, the equation is not solvable in radicals.