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

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- May 17th 2009, 12:50 PMZetaXGalois Group
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 - May 17th 2009, 01:33 PMThePerfectHacker
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.