Editing Galois theory (section)

==Solvable groups and solution by radicals==

The notion of a [[solvable group]] in [[group theory]] allows one to determine whether a polynomial is solvable in radicals, depending on whether its Galois group has the property of solvability. In essence, each field extension {{math|''L''/''K''}} corresponds to a [[factor group]] in a [[composition series]] of the Galois group. If a factor group in the composition series is [[cyclic group|cyclic]] of order {{math|''n''}}, and if in the corresponding field extension {{math|''L''/''K''}} the field {{math|''K''}} already  contains a [[Root of unity|primitive {{math|''n''}}th root of unity]], then it is a radical extension and the elements of {{math|''L''}} can then be expressed using the {{math|''n''}}th root of some element of {{math|''K''}}.

If all the factor groups in its composition series are cyclic, the Galois group is called ''solvable'', and all of the elements of the corresponding field can be found by repeatedly taking roots, products, and sums of elements from the base field (usually {{math|'''Q'''}}).

One of the great triumphs of Galois Theory was the proof that for every {{math|''n'' > 4}}, there exist polynomials of degree {{math|''n''}} which are not solvable by radicals (this was proven independently, using a similar method, by [[Niels Henrik Abel]] a few years before, and is the [[Abel–Ruffini theorem]]), and a systematic way for testing whether a specific polynomial is solvable by radicals. The Abel–Ruffini theorem results from the fact that for {{math|''n'' > 4}} the [[symmetric group]] {{math|''S''<sub>''n''</sub>}} contains a [[simple group|simple]], noncyclic, [[normal subgroup]], namely the [[alternating group]] {{math|''A''<sub>''n''</sub>}}.

==={{anchor|Abel-Ruffini theorem}}A non-solvable quintic example===
[[image:Non solvable quintic.svg|right|thumb|For the polynomial {{math|''f''(''x'') {{=}} ''x''<sup>5</sup> − ''x'' − 1}}, the lone real root {{math|''x'' {{=}} 1.1673...}} is algebraic, but not expressible in terms of radicals. The other four roots are [[complex numbers]].]]

[[Bartel Leendert van der Waerden|Van der Waerden]]<ref>van der Waerden, Modern Algebra (1949 English edn.), Vol. 1, Section 61, p.191<!-- p. 191 isn't in the preview, but this instance might be a better ref. {{cite book |first=B.L. |last=van der Waerden |first2=Emil |last2=Artin |first3=Emmy |last3=Noether |title=Algebra: Based in Part on Lectures by E. Artin and E. Noether. ... |url=https://books.google.com/books?id=XDN8yR8R1OUC&pg=PP1 |date=2003 |publisher=Springer |isbn=978-0-387-40624-4 |volume=1 |origyear=1970}} --></ref> cites the polynomial {{math|''f''(''x'') {{=}} ''x''<sup>5</sup> − ''x'' − 1}}.  

By the [[rational root theorem]], this has no rational zeroes. 

Neither does it have linear factors modulo 2 or 3.

The Galois group of {{math|''f''(''x'')}} modulo 2 is cyclic of order 6, because {{math|''f''(''x'')}} modulo 2 factors into polynomials of orders 2 and 3, {{math|(''x''<sup>2</sup> + ''x'' + 1)(''x''<sup>3</sup> + ''x''<sup>2</sup> + 1)}}.

{{math|''f''(''x'')}} modulo 3 has no linear or quadratic factor, and hence is irreducible.  Thus its modulo 3 Galois group contains an element of order 5.

It is known<ref>{{cite book |first=V.V. |last=Prasolov |chapter=5 Galois Theory Theorem 5.4.5(a) |doi=10.1007/978-3-642-03980-5_5 |title=Polynomials |publisher=Springer |series=Algorithms and Computation in Mathematics |volume=11 |year=2004 |isbn=978-3-642-03979-9 |pages=181–218 }}</ref> that a Galois group modulo a prime is isomorphic to a subgroup of the Galois group over the rationals.  A permutation group on 5 objects with elements of orders 6 and 5 must be the symmetric group {{math|''S''<sub>5</sub>}}, which is therefore the Galois group of {{math|''f''(''x'')}}. 

This is one of the simplest examples of a non-solvable quintic polynomial.  According to [[Serge Lang]], [[Emil Artin]] was fond of this example.<ref>{{cite book |title=Algebraic Number Theory|volume=110|series=Graduate Texts in Mathematics|first=Serge|last=Lang|author-link=Serge Lang|publisher=Springer|year=1994|isbn=9780387942254|page=121|url=https://books.google.com/books?id=u5eGtA0YalgC&pg=PA}}</ref>