Answer to Question #136059 in Abstract Algebra for Nabanita

Question #136059
What is the total number of subgroup of the galois group of the polynomial x^3-2 over Q?
1
Expert's answer
2020-10-01T15:05:47-0400

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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