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Math Help - Question concerning decomposition

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    Question concerning decomposition

    Hello

    two questions:

    a) Write \mathbb{F}_9[y] \diagup (1+y+y^3+y^4) as a direct sum.
    b) Why doesn't exist a direct sum decomposition of \mathbb{F}_{5^{2744}}[y] \diagup (y^{160675})?


    a) Since f(y)=1+y+y^3+y^4 = (1+y)*(1+y^3)=h(y)*g(y) the roots of f are contained in the field considered. y is a root of f if it's a root of h or it is a root of g. So my decomposition here is

    \mathbb{F}_9[y] \diagup (1+y)  \oplus    \mathbb{F}_9[y] \diagup (1+y^3)

    Is it correct and what about my reason? 9=3^2 but why can that not be decomposed more?


    b) 2744=2^3*7^3
    160675=5^2*6427

    Are the numbers choosed with this factorisation for causing confusion only? My reason here is:

    If y is a root of f(y)=y^{160675} it is not a root of a polynomial dividing f(y) of a lower degree. No matter what y^{160674} is the ys contained in the field considered at b) are only 0 when powered to 160675.


    What do you think of my explanations? How would you reason here?

    Regards
    Last edited by huberscher; October 11th 2013 at 11:35 AM.
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