Find the galois group of
I multiply the polynomial with 24 then get
Then I have that the cubic resolvent of is
Therefore, I get that the galois group of is
My question is: Can I so conclude that the Galois group of
is
Let then and have the same Galois group. Because the splitting field of and are identical. The Galois group of a polynomial is defined to be the Galois group of the splitting field over the base field, since the splitting fields are the same it follows that the Galois groups are the same.