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Wikipedia: Abel–Ruffini theorem
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The following proof is based on Galois 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 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 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]. 

Multiplying 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}. Because our independent transcendentals y_n act as indeterminates over Q, every permutation \sigma in the symmetric group on 5 letters S_5 induces an 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 E fixed, so they are elements of the Galois group G(E/Q). Now, since |S_5| = 5! it must be that |G(E/Q)| \ge 5!, as there could possibly be automorphisms there that are not in S_5. However, since the relative automorphisms Q for splitting field of a quintic polynomial has at most 5! elements, |G(E/Q)| = 5!, and so G(E/Q) must be 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. 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

Objev důkazu nebyl snadný:

  • 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. 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.[3]
  • 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.
  • In 1963, Vladimir Arnold discovered a topological proof of the Abel–Ruffini theorem,[4] which served as a starting point for topological Galois theory.[5]