Editing Quaternion group (section)

==Galois group==
[[Richard Dedekind]] considered the field <math>\mathbb{Q}[\sqrt{2}, \sqrt{3}]</math> in attempting to relate the quaternion group to [[Galois theory]].<ref>[[Richard Dedekind]] (1887) "Konstrucktion der Quaternionkörpern", Ges. math. Werk II 376–84</ref> In 1936 [[Ernst Witt]] published his approach to the quaternion group through Galois theory.<ref>[[Ernst Witt]] (1936) "Konstruktion von galoisschen Körpern..."[[Crelle's Journal]] 174: 237-45</ref>

In 1981, Richard Dean showed the quaternion group can be realized as the [[Galois group]] Gal(T/'''Q''') where '''Q''' is the field of [[rational number]]s and T is the [[splitting field]] of the polynomial
:<math>x^8 - 72 x^6 + 180 x^4 - 144 x^2 + 36</math>.
The development uses the [[fundamental theorem of Galois theory]] in specifying four intermediate fields between '''Q''' and T and their Galois groups, as well as two theorems on cyclic extension of degree four over a field.<ref name=":0">{{cite journal
 | last = Dean | first =  Richard
 | year = 1981
 | title = A Rational Polynomial whose Group is the Quaternions
 | journal = [[American Mathematical Monthly|The American Mathematical Monthly]]
 | volume = 88 | issue = 1 | pages = 42–45
 | doi = 10.2307/2320711
 | jstor = 2320711
}}</ref>