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
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