Find the galois group of $\displaystyle f(x) =1+x+x^2/2+x^3/6+x^4/24.$

I multiply the polynomial with 24 then get

$\displaystyle 24f(x) =24+24x+12x^2+4x^3+x^4.$

Then I have that the cubic resolvent of $\displaystyle 24f(x)$ is

$\displaystyle h(x) = z^3-12z^2+192.$

Therefore, I get that the galois group of $\displaystyle 24f(x)$ is $\displaystyle A_4.$

My question is: Can I so conclude that the Galois group of $\displaystyle f(x)$

is $\displaystyle A_4.$