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Thread: Splitting fields

  1. #1
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    Splitting fields

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

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

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

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

    2)$\displaystyle x^{4}-7x^{2}+10$

    splitting field $\displaystyle (x^{2}-5)(x^{2}-2)$

    [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)$\displaystyle X^{4}-5$
    I think i understand just need reasurance

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

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


    But you haven't yet determined $\displaystyle K$ , and its degree is wrong: certainly $\displaystyle [\mathbb{Q}(\sqrt[4]{5}):\mathbb{Q}]=4$ , 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, $\displaystyle \sqrt[4]{5}\,i$ is a root of the pol. that isn't contained in $\displaystyle \mathbb{Q}(\sqrt[4]{5})$.

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

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

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


    2)$\displaystyle x^{4}-7x^{2}+10$

    splitting field $\displaystyle (x^{2}-5)(x^{2}-2)$

    [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 $\displaystyle K$ ...

    Tonio


    thanks bobisback
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