Editing Galois theory (section)

===Quartic equation===
Consider the polynomial
:<math>x^4 - 10x^2 + 1.</math>
[[Completing the square]] in an unusual way, it can also be written as
:<math>(x^2-1)^2-8x^2 = (x^2-1-2x\sqrt2 )(x^2-1+2x\sqrt 2).</math>
By applying the [[quadratic formula]] to each factor, one sees that the four roots are  
:<math>\begin{align}
A &= \sqrt{2} + \sqrt{3},\\
B &= \sqrt{2} - \sqrt{3},\\
C &= -\sqrt{2} + \sqrt{3},\\
D &= -\sqrt{2} - \sqrt{3}.
\end{align}</math>

Among the 24 possible [[permutation]]s of these four roots, four are particularly simple, those consisting in the sign change  of 0, 1, or 2 square roots. They form a group that is isomorphic to the [[Klein four-group]]. 

Galois theory implies that, since the polynomial is irreducible, the Galois group has at least four elements. For proving that the Galois group consists of these four permutations, it suffices thus to show that every element of the Galois group is determined by the image of {{mvar|A}}, which can be shown as follows.

The members of the Galois group must preserve any algebraic equation with rational coefficients involving {{math|''A''}}, {{math|''B''}}, {{math|''C''}} and {{math|''D''}}.

Among these equations, we have:
:<math>\begin{align}
AB&=-1 \\ AC&=1 \\ A+D&=0
\end{align}</math>

It follows that, if {{math|''φ''}} is a permutation that belongs to the Galois group, we must have:
:<math>\begin{align}
\varphi(B)&=\frac{-1}{\varphi(A)}, \\ \varphi(C)&=\frac{1}{\varphi(A)}, \\ \varphi(D)&=-\varphi(A).
\end{align}</math>
This implies that the permutation is well defined by the image of {{math|''A''}}, and that the Galois group has 4 elements, which are:
:{{math|(''A'', ''B'', ''C'', ''D'') → (''A'', ''B'', ''C'', ''D'')}}{{spaces|5}}(identity)
:{{math|(''A'', ''B'', ''C'', ''D'') → (''B'', ''A'', ''D'', ''C'')}}{{spaces|5}}(change of sign of <math>\sqrt3</math>)
:{{math|(''A'', ''B'', ''C'', ''D'') → (''C'', ''D'', ''A'', ''B'')}}{{spaces|5}}(change of sign of <math>\sqrt2</math>)
:{{math|(''A'', ''B'', ''C'', ''D'') → (''D'', ''C'', ''B'', ''A'')}}{{spaces|5}}(change of sign of both square roots)
This implies that the Galois group is isomorphic to the [[Klein four-group]].