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Finding subfields

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Find all subfields of Q(sqrt(2), sqrt(3))with proof that you have them all. What is the minimal polynomial of sqrt(2) + sqrt(3)? Which of your subfields does it generate over Q?

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This provides an example of finding all subfields and proving they were all found.

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Let K = Q (sqrt 2, sqrt 3); then K is a normal extension of Q of degree 4, with Galois group isomorphic to the Klein 4-group.

Indeed, the automorphisms of K which leave Q fixed are given below by the values on the basis
{ 1, sqrt 2, sqrt 3, sqrt 6} for K/Q:

1, the identity,

s_1 : sqrt 2 |--> - sqrt 2; sqrt 6 |--> - sqrt 6; leaves the other fixed

s_2 : sqrt 3 |--> - sqrt 3; sqrt 6 |--> - sqrt 6; leaves the others fixed

s_3 : sqrt 2 |--> - sqrt 2; sqrt 3 |--> - sqrt 3; leaves the ...

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