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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?

    © BrainMass Inc. brainmass.com October 9, 2019, 10:25 pm ad1c9bdddf
    https://brainmass.com/math/basic-algebra/finding-subfields-219636

    Solution Preview

    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 ...

    Solution Summary

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

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