= Split resolvent criterion for a Klein four Galois group
{c}
Let a <separable polynomial> $f$ of degree four be irreducible over a <field> $F$. If all three pair sums $r_1=\alpha_1\alpha_2+\alpha_3\alpha_4$, $r_2=\alpha_1\alpha_3+\alpha_2\alpha_4$, and $r_3=\alpha_1\alpha_4+\alpha_2\alpha_3$ lie in $F$, its <Galois group> is the <Klein four-group>. Indeed $r_1-r_2=(\alpha_1-\alpha_4)(\alpha_2-\alpha_3)\ne0$, and similarly the other pair sums differ. Every <Galois automorphism> therefore fixes each of the three partitions into pairs. The kernel of this action of $S_4$ on three partitions is $V_4=\{1,(12)(34),(13)(24),(14)(23)\}$. Irreducibility gives a <transitive group action> on the four roots, so the <orbit-stabilizer theorem> forces $4$ to divide the group order. Hence the <Galois group> equals $V_4$. The argument also works in characteristic two.
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