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Math Help - normal extension of the rationals - quick questions

  1. #1
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    normal extension of the rationals - quick questions

    Just trying to get my head around normal extensions.

    True or False?

    is a normal extension of

    is a normal extension of

    I would say for the first one, true, because it is the splitting field of .

    For the second one, false, because is irreducible by Eisenstein, and has only one zero > 0 in

    Could anyone please tell me if these are OK?

    Thanks in advance!
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  2. #2
    Super Member Gamma's Avatar
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    Bingo

    Looks good to me.
    You can tell the second is not normal because the other two roots are imaginary and \mathbb{Q}(2^{1/3})\subset \mathbb{R}. But yeah I think you got it.
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