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Consider the polynomial equation \begin{equation*} x^4 - 2 = 0, \end{equation*} with integer coefficients. We have four solutions: \begin{equation*} \alpha = \sqrt[4]{2},\quad \beta = i \sqrt[4]{2},\quad \gamma = -\sqrt[4]{2},\quad \delta = -i \sqrt[4]{2}, \end{equation*} Note that two of these solutions are not real. Now let us list some equations that these four roots satisfy: \begin{equation*} \alpha + \gamma = 0,\quad \alpha \beta \gamma \delta = -2,\quad \alpha \beta - \gamma \delta = 0, \dotsc \end{equation*}

What happens if we swap $ \alpha $ and $ \gamma $ in this list? We get \begin{equation*} \gamma + \alpha = 0, \quad \gamma \beta \alpha \delta = -2,\quad \gamma \beta - \alpha \delta = 0, \dotsc \end{equation*} which are all still valid. Likewise we could permute the variables as follows \begin{equation*} \alpha \mapsto \beta \mapsto \gamma \mapsto \delta \end{equation*} You can check that the equations above remain valid. So, does any such permutation work? For any such equation? The answer is "no": try, for instance, swapping $ \beta $ and $ \gamma $, and you find that the first of the equations above becomes false.

The set of permutations of the set $ \{\alpha, \beta, \gamma, \delta \} $ that preserve the validity of all polynomial equations (with coefficients in $ \mathbb{Q} $) in these variables is called the Galois group of the equation $ x^4 = 2 $. This group is the symmetric group of the square (which makes sense if you look at the location of $ \alpha, \beta, \gamma $, and $ \delta $ in the Complex plane. It is also known as $ D_8 $, the dihedral group of order 8.

Question 1: In order to determine the permutations of the set $ \{ \alpha, \beta, \gamma, \delta \} $ that preserve the validity of all polynomial equations (with coefficients in Q) in these variables, do I need to list out all such equations (how would I do it), and how do tell the list is exhaustive and then only determine such permutations?

Question 2: Following Question 1, the coefficients of a polynomial form symmetric polynomials as functions of the roots. How this help in determining the correct permutations? Using this as example, the Galois group is isomorphic to the Klein 4-group: $ x^4 - 5x + 6 = 0 $ factorize as $ (x^2 - 3)(x^2 - 2) = (x^2 - r_1 r_2)(x^2 - r_3 r_4) $ and the permutation that leave the polynomial unchanged is the identity, $ (12), (34) $ and $ (12)(34) $. Other permutations turn it into different polynomials and do not belong the Galois group of the permutation?

Question 3: Am I correct to say that a permutation belongs to the Galois group of the polynomial if it permutes the roots in such a way that when we multiply out, we obtain the polynomial that we started with?

Question 4: How would Galois himself have determined the "Galois group" of a given polynomial?

Question 5: Could you please point out where I could find materials on how Galois solved the insolvability of quintic using materials available during his time, instead of the automorphism of splitting field approach? My google search has been unsatisfatory. The textbooks by Cox, Tignol, Rotman, Swallow are not easy to me and do not have the discussions I am comfortable with.


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