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Math Help - finite fields

  1. #1
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    finite fields

    Let F=(Z₃[x])/<x+1>
    Calculate (x + 1)⁴ in F. Explain why your calculation shows that F* is cyclic

    I have calculated (x + 1)⁴and then divided the result by x^2+1 which gives me x+2

    I am not sure if i have done this correctly and i dont see how this shows me that the multiplicative group will be cyclic.
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  2. #2
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    Quote Originally Posted by ulysses123 View Post
    Let F=(Z₃[x])/<x+1>
    Calculate (x + 1)⁴ in F. Explain why your calculation shows that F* is cyclic

    I have calculated (x + 1)⁴and then divided the result by x^2+1 which gives me x+2

    I am not sure if i have done this correctly and i dont see how this shows me that the multiplicative group will be cyclic.

    It seems you wrote F_9:=Z_3[x]/<x^2+1>= the field with 9 elements.
    Now , I don't know what is (x+1)' or whatever it was you wrote, but perhaps the intention was to

    calculate the powers of (x+1)+<x^2+1> \in F_9 and, thus, show that this is an

    element of order 8, thus showing that F_9^* is a cyclic group...
    But then I don't understand why did you divide x+1\,\,by\,\,x^2+1...??

    Explain better your symbols and what exactly have you done.

    Anyway, it's easy to check that [(x+1)+<x^2+1>]^4=-1 and thus x+1 indeed has order 8 in the multiplicative group of the field.

    Tonio
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