Editing Galois theory (section)

==={{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>