Galois group of a degree 5 irreducible polynomial with two complex roots.

group-theory galois-theory

Solution 1:

Let the roots be $a,b,c,d,e$ with $a,b $ non-real. Your subgroup is transitive and has at least one transposition $(a b)$. Since it is transitive you can conjugate your transposition by some element to get a transposition which moves $c$ (conjugation preserves the cycle structure), and likewise one which moves $d$ or $e$.

Case 1. If this is $(bc)$ or $(ac)$ you can find your 3-cycle as a product of transpositions.

Case 2. If it is $(cd)$ then you can use conjugation to find a transposition which moves $e$ and whichever one you get, you can find your 3-cycle by composing it with either $(ab)$ or $(cd)$.

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