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Math Help - Another field contruction

  1. #1
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    Another field contruction

    Construct a field of order 125.

    I don't know, this chapter is just not coming to me for some reasons, I can't really understand how to construct a field. So I pick a field, then pick a poly that is irreducible and has order of 125, how do I do that?
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Construct a field of order 125.

    I don't know, this chapter is just not coming to me for some reasons, I can't really understand how to construct a field. So I pick a field, then pick a poly that is irreducible and has order of 125, how do I do that?
    Again remember what I said. Let F be a finite field \mathbb{Z}_p. Let p(x) = a_nx^n+...+a_1x+a_0 be an irreducible polynomial over F. Then F/\left< p(x) \right> is a field. Now any element in F/\left< p(x) \right> can be written uniquely as b_0+b_1x+...+b_{n-1}x^{n-1} now for b_i (for 0\leq i \leq n-1) we have p choices for the coefficients. Thus, in total there are p^n such elements in this larger field.

    1)Let F=\mathbb{Z}_5
    2)Let p(x) be an order 3 irreducible polynomial in F.
    3)The factor ring F/\left< p(x)\right> is a field with 5^3=125 elements.

    So the thing remaining now is for you to find an irreducible degree polynomial.
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  3. #3
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    (x-2)^3, it has zero in there. would that be it?
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  4. #4
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    Quote Originally Posted by tttcomrader View Post
    (x-2)^3, it has zero in there. would that be it?
    I am not sure what you are asking. Do you follow what I did? You need to find and irreducible polynomial of degree 3 over Z_5.
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