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Math Help - Splitting fields

  1. #1
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    Splitting fields

    determine the splitting field K/Q and calculate [K:Q]

    i) X^{4}-5
    I think i understand just need reasurance

    spliting field (x^{2}-\sqrt{5})(x^{2}+\sqrt{5})

    [K:Q] =4 because x^{4} - 5 is min polynomial

    2) x^{4}-7x^{2}+10

    splitting field (x^{2}-5)(x^{2}-2)

    [K:Q] = 4

    is this all correct?

    thanks bobisback
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  2. #2
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    Quote Originally Posted by bobisback View Post
    determine the splitting field K/Q and calculate [K:Q]




    i) X^{4}-5
    I think i understand just need reasurance

    spliting field (x^{2}-\sqrt{5})(x^{2}+\sqrt{5})

    [K:Q] =4 because x^{4} - 5 is min polynomial


    But you haven't yet determined K , and its degree is wrong: certainly [\mathbb{Q}(\sqrt[4]{5}):\mathbb{Q}]=4 , but this extension field of the rationals is NOT

    the splitting field of the given pol. over the rationals since it doesn't contain ALL its roots...for example, \sqrt[4]{5}\,i is a root of the pol. that isn't contained in \mathbb{Q}(\sqrt[4]{5}).

    What you did is just fine to find the roots: x^4-5=(x^2-\sqrt{5})(x^2+\sqrt{5}) , but then it must be clear that both roots of the rightmost factor are complex

    non-real ( in fact, a conjugate pair: \pm \sqrt[4]{5}\,i ) , so they aren't contained in a real field as \mathbb{Q}(\sqrt[4]{5}) .

    Take it from here and try now to find explicitly what K is and its degree over the rationals.


    2) x^{4}-7x^{2}+10

    splitting field (x^{2}-5)(x^{2}-2)

    [K:Q] = 4

    is this all correct?


    This time the degree is correct (though you gave no explanation at all... ) but again you haven't explicitly described the splitting field K ...

    Tonio


    thanks bobisback
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