Editing Abel–Ruffini theorem (section)

==Context==
[[Polynomial equation]]s of degree two can be solved with the [[quadratic formula]], which has been known since [[ancient history|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 [[real number|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 [[polynomial root-finding|several methods]] are known for  approximating all solutions to any desired accuracy.

From the 16th century to 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 (mathematics)|expression]] involving only the coefficients of the equation, and the operations of [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[nth root extraction|{{mvar|n}}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 <math>x^n-1=0</math> for any {{mvar|n}}, and the equations defined by [[cyclotomic polynomial]]s, 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."<ref name="Stewart">{{Citation|last=Stewart|first=Ian|author-link=Ian Stewart (mathematician)|title=Galois Theory|chapter=Historical Introduction|publisher=[[CRC Press]]|isbn=978-1-4822-4582-0|year=2015|edition=4th}}</ref> 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 computer]]s. With modern computers and programs, deciding whether a polynomial is solvable by radicals can be done for polynomials of degree greater than 100.<ref>{{citation
 | last1 = Fieker | first1 = Claus
 | last2 = Klüners | first2 = Jürgen
 | doi = 10.1112/S1461157013000302
 | issue = 1
 | journal = LMS Journal of Computation and Mathematics
 | mr = 3230862
 | pages = 141–158
 | title = Computation of Galois groups of rational polynomials
 | volume = 17
 | year = 2014| arxiv = 1211.3588
 }}</ref> 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.