Editing Galois theory (section)

==Permutation group approach==<!-- [[Galois group of a polynomial]] redirects here-->
{{Unreferenced section|date=June 2023}}

Given a polynomial, it may be that some of the roots are connected by various [[algebraic equation]]s. For example, it may be that for two of the roots, say {{math|''A''}} and {{math|''B''}}, that {{math|1=''A''<sup>2</sup> + 5''B''<sup>3</sup> = 7}}. The central idea of Galois' theory is to consider [[permutation]]s (or rearrangements) of the roots such that ''any'' algebraic equation satisfied by the roots is ''still satisfied'' after the roots have been permuted. Originally, the theory had been developed for algebraic equations whose coefficients are [[rational number]]s. It extends naturally to equations with coefficients in any [[field (mathematics)|field]], but this will not be considered in the simple examples below.<ref>"Lecture Notes on Galois Theory". ''UC Davis''.https://www.math.ucdavis.edu/~egorskiy/MAT150C-s17/Galois.pdf</ref>

These permutations together form a [[permutation group]], also called the [[Galois group]] of the polynomial, which is explicitly described in the following examples.

===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.

===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]].