Editing Cubic equation (section)

==Galois group==
Given a cubic [[irreducible polynomial]] over a field {{mvar|K}} of [[characteristic (algebra)|characteristic]] different from 2 and 3, the [[Galois theory#Permutation group approach to Galois theory|Galois group]] over {{mvar|K}} is the group of the [[field automorphism]]s that fix {{mvar|K}} of the smallest extension of {{mvar|K}} ([[splitting field]]). As these automorphisms must permute the roots of the polynomials, this group is either the group {{math|''S''<sub>3</sub>}} of all six permutations of the three roots, or the group {{math|''A''<sub>3</sub>}} of the three circular permutations.

The discriminant {{math|Δ}} of the cubic is the square of 
<math display="block">\sqrt \Delta =a^2(r_1-r_2)(r_1-r_3)(r_2-r_3),</math>
where {{mvar|a}} is the leading coefficient of the cubic, and {{math|''r''<sub>1</sub>}}, {{math|''r''<sub>2</sub>}} and {{math|''r''<sub>3</sub>}} are the three roots of the cubic. As <math>\sqrt \Delta</math> changes of sign if two roots are exchanged, <math>\sqrt \Delta</math> is fixed by the Galois group only if the Galois group is 
{{math|''A''<sub>3</sub>}}. In other words, the Galois group is {{math|''A''<sub>3</sub>}} if and only if the discriminant is the square of an element of {{mvar|K}}.

As most integers are not squares, when working over the field {{math|'''Q'''}} of the [[rational number]]s, the Galois group of most irreducible cubic polynomials is the group {{math|''S''<sub>3</sub>}} with six elements. An example of a Galois group {{math|''A''<sub>3</sub>}} with three elements is given by {{math|''p''(''x'') {{=}} ''x''<sup>3</sup> − 3''x'' − 1}}, whose discriminant is {{math|81 {{=}} 9<sup>2</sup>}}.