Editing Galois theory (section)

===Quadratic equation===
Consider the [[quadratic equation]]
:<math>x^2 - 4x + 1 = 0. </math>
By using the [[Quadratic equation#Quadratic formula and its derivation|quadratic formula]], we find that the two roots are
:<math>\begin{align}
A &= 2 + \sqrt{3},\\
B &= 2 - \sqrt{3}.
\end{align}</math>
Examples of algebraic equations satisfied by {{math|''A''}} and {{math|''B''}} include
:<math>A + B = 4, </math>
and
:<math>AB = 1. </math>

If we exchange {{math|''A''}} and {{math|''B''}} in either of the last two equations we obtain another true statement. For example, the equation {{math|''A'' + ''B'' {{=}} 4}} becomes {{math|''B'' + ''A'' {{=}} 4}}. It is more generally true that this holds for ''every'' possible [[algebraic equation|algebraic relation]] between {{math|''A''}} and {{math|''B''}} such that all [[coefficients]] are [[rational number|rational]]; that is, in any such relation, swapping {{math|''A''}} and {{math|''B''}} yields another true relation. This results from the theory of [[symmetric polynomial]]s, which, in this case, may be replaced by formula manipulations involving the [[binomial theorem]].

One might object that {{math|''A''}} and {{math|''B''}} are related by the algebraic equation {{math|''A'' − ''B'' − 2{{sqrt|3}} {{=}} 0}}, which does not remain true when {{math|''A''}} and {{math|''B''}} are exchanged. However, this relation is not considered here, because it has the coefficient {{math|−2{{sqrt|3}}}} which is [[Quadratic irrational#Square root of non-square is irrational|not rational]].

We conclude that the Galois group of the polynomial {{math|''x''<sup>2</sup> − 4''x'' + 1}} consists of two permutations: the [[Permutation#Circular permutations|identity]] permutation which leaves {{math|''A''}} and {{math|''B''}} untouched, and the [[transposition (mathematics)|transposition]] permutation which exchanges {{math|''A''}} and {{math|''B''}}. As all groups with two elements are [[group isomorphism|isomorphic]], this Galois group is isomorphic to the [[multiplicative group]] {{math|{{mset|1, −1}}}}.

A similar discussion applies to any quadratic polynomial {{math|''ax''<sup>2</sup> + ''bx'' + ''c''}}, where {{math|''a''}}, {{math|''b''}} and {{math|''c''}} are rational numbers.
* If the polynomial has rational roots, for example {{math|''x''<sup>2</sup> − 4''x'' + 4 {{=}} (''x'' − 2)<sup>2</sup>}}, or {{math|''x''<sup>2</sup> − 3''x'' + 2 {{=}} (''x'' − 2)(''x'' − 1)}}, then the Galois group is trivial; that is, it contains only the identity permutation. In this example, if {{math|''A'' {{=}} 2}} and {{math|''B'' {{=}} 1}} then {{math|''A'' − ''B'' {{=}} 1}} is no longer true when {{math|''A''}} and {{math|''B''}} are swapped.
* If it has two [[irrational number|irrational]] roots, for example {{math|''x''<sup>2</sup> − 2}}, then the Galois group contains two permutations, just as in the above example.