# Solvability of polynomial equations of degree 4 (see question at the end)

Let $f$ be irreducible and separable over $k$ and $\text{char}(k) \neq 2, 3$. Further let $f(x) = x^4 +px^2 +qx + r = (x-\alpha_1)(x-\alpha_2)(x-\alpha_3)(x-\alpha_4)$ which splits over $K$.

Then:

$G:=\text{Gal}(f)\hookrightarrow S_4$

$G\cap V_4$ is a normal subgroup of $G$, with $V_4 = \{\text{id},(12)(34),(13)(24),(14)(23)\}$

Now, the situation is as follows:

$k$ --- $K^{G\cap V_4}$ --- $K$

Letting $\beta := \alpha_1 + \alpha_2, \;\gamma := \alpha_1 + \alpha_3, \;\delta := \alpha_1 + \alpha_4$ we get (considering that $\alpha_1 + \alpha_2 +\alpha_3 + \alpha_4 = 0$):

$\beta^2 = (\alpha_1 + \alpha_2)^2 = -(\alpha_1 + \alpha_2)(\alpha_3 + \alpha_4)$

$\gamma^2 = (\alpha_1 + \alpha_3)^2 = -(\alpha_1 + \alpha_3)(\alpha_2 + \alpha_4)$

$\delta^2 = (\alpha_1 + \alpha_4)^2 = -(\alpha_1 + \alpha_4)(\alpha_2 + \alpha_3)$

$\Rightarrow \sigma(\beta^2)=\beta^2, \sigma(\gamma^2)=\gamma^2, \sigma(\delta^2)=\delta^2\;\; \forall \sigma \in G\cap V_4$

$\Rightarrow k(\beta^2, \gamma^2, \delta^2)\subseteq K^{G\cap V_4}$

Now,

$\alpha_1 = \frac{1}{2}(\beta + \gamma + \delta)$

$\alpha_2 = \frac{1}{2}(\beta - \gamma - \delta)$

$\alpha_3 = \frac{1}{2}(-\beta + \gamma - \delta)$

$\alpha_4 = \frac{1}{2}(-\beta - \gamma + \delta)$

$\Rightarrow k(\beta^2, \gamma^2, \delta^2)(\beta, \gamma, \delta) = k(\beta, \gamma, \delta) = k(\alpha_1, \alpha_2, \alpha_3) = K$

Here comes the part I don't understand:

$\Rightarrow \sigma \in \text{Gal}(K/k(\beta^2, \gamma^2, \delta^2)) \Rightarrow \sigma^2 = \text{id}$

Why does $\sigma \in \text{Gal}(K/k(\beta^2, \gamma^2, \delta^2))$ imply $\sigma^2 = \text{id}$?

If $\sigma \in Gal(K/k(\beta^2,\gamma^2,\delta^2))$, then by definition $\sigma(\beta^2)=\beta^2$, $\sigma(\gamma^2)=\gamma^2$, and $\sigma(\delta^2)=\delta^2$, because $\sigma$ fixes $k(\beta^2,\gamma^2,\delta^2)$. Now look at the expressions you listed for $\beta^2$, $\gamma^2$, and $\delta^2$, $$\beta^2=-(\alpha_1 + \alpha_2)(\alpha_3 + \alpha_4)$$ $$\gamma^2=-(\alpha_1 + \alpha_3)(\alpha_2 + \alpha_4)$$ $$\delta^2=-(\alpha_1 + \alpha_4)(\alpha_2 + \alpha_3)$$
and consider the effect of any permutation $\sigma'$ not in $V_4$. In particular, observe that two of the equalities $\sigma'(\beta^2)=\beta^2$, $\sigma'(\gamma^2)=\gamma^2$, $\sigma'(\delta^2)=\delta^2$ fail to hold. This implies that $Gal(K/k(\beta^2,\gamma^2,\delta^2)) = G\cap V_4$ and $\sigma^2=$ id. Hope this helps!