# Finding Galois group, its Lattice and correspondence

I am trying to find the splitting field $L$ of $x^3-8 \in \mathbb Q[x]$ and the degree of the extension $\mathbb Q \subset L$, computing Gal$(L/\mathbb Q)$ and finding the correspondence between the subgroups of it and the intermediate field extensions. Also I have to show that which intermediate extensions are normal in $\mathbb Q$.

I tried to solve it, the spliiting field is $\mathbb Q(i\sqrt3)$ and the degree of the extension is 2. How to proceed further? Specially about the correspondence?

1 and $i\sqrt3$ forms a basis of $L$ over $\mathbb Q$. so every element of $L$ has the form $a+bi\sqrt3$ If $f \in Gal(L/\mathbb Q)$ then $f(a+bi\sqrt3)=a+bf(i\sqrt 3)$ Now $(f(i\sqrt3))^2=f(-3)=-3 \implies f(i\sqrt3)=i\sqrt3$ Is it correct and the Galois group is ?

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Is the Galois group trivial or $\mathbb Z_2$ –  Mathematician Feb 11 at 2:44
Being this a Galois extension the Galois group must have order equal to the extension's degree: $\,2\,$, but check that $\,f(i\sqrt 3)^2=f(-3)=-3\Longrightarrow f(i\sqrt 3)=\pm\,i\sqrt 3\,$ , and you have two choices... –  DonAntonio Feb 11 at 3:26
The problem would be much more interesting if it were $x^3-2$. –  Gerry Myerson Feb 11 at 3:53
yes it is $\mathbb Z_2$ –  Mathematician Feb 11 at 4:40