Editing Galois group (section)

=== Finite non-abelian groups ===
Consider now <math>L = \Q(\sqrt[3]{2}, \omega),</math> where <math>\omega</math> is a [[root of unity|primitive cube root of unity]]. The group <math>\operatorname{Gal}(L/\Q)</math> is isomorphic to {{math|''S''<sub>3</sub>}}, the [[dihedral group of order 6]], and {{math|''L''}} is in fact the splitting field of <math>x^3-2</math> over <math>\Q.</math>

==== Quaternion group ====
The [[Quaternion group]] can be found as the Galois group of a field extension of <math>\Q</math>. For example, the field extension

:<math>\Q \left (\sqrt{2}, \sqrt{3}, \sqrt{(2+\sqrt{2})(3+\sqrt{3})} \right )</math>

has the prescribed Galois group.<ref>{{Cite book| last=Milne| url= https://www.jmilne.org/math/CourseNotes/ft.html| title=Field Theory|pages=46}}</ref>

==== Symmetric group of prime order ====
If <math>f</math> is an [[irreducible polynomial]] of prime degree <math>p</math> with rational coefficients and exactly two non-real roots, then the Galois group of <math>f</math> is the full [[symmetric group]] <math>S_p.</math><ref name=":1">{{Cite book| last=Lang| first=Serge| title=Algebra| edition=Revised Third| pages=263, 273}}</ref>

For example, <math>f(x)=x^5-4x+2 \in \Q[x]</math> is irreducible from Eisenstein's criterion. Plotting the graph of <math>f</math> with graphing software or paper shows it has three real roots, hence two complex roots, showing its Galois group is <math>S_5</math>.