Question

What is the total number of subgroup of the galois group of the polynomial x^3-2 over Q?

Answer 1

The Galois group of the polynomial $x^3-2$ over $\mathbb Q$ is isomorphic to the symmetric group $S_3=\{(1),(12),(13),(23),(123),(132)\}.$ By the Lagrange's theorem, for any finite group $S_3$, the order of every subgroup $H$ of $S_3$ divides the order of $S_3$, that is $|H|\in\{1,2,3,6\}$. All groups of prime orders are cyclic.

Let us find all subgroup of $S_3$:

$\langle(12)\rangle=\{(1),(12)\}, \langle(13)\rangle=\{(1),(13)\}, \langle(23)\rangle=\{(1),(23)\}, \langle(123)\rangle=\{(1),(123),(132)\}=\langle(132)\rangle, \{(1)\}, S_3.$

It follows that the total number of subgroups the Galois group of the polynomial $x^3-2$ over $\mathbb Q$

is six.

Answer: 6