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Math Help - field of order p^2

  1. #1
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    field of order p^2

    Hi again!
    My problem is to show that a field of order p^2 exists for every prime p.

    In an earlier problem I found that there were p^2monic quadratics in  Z_p[x], but I don't know if that's useful. I also showed that (1/2)(p^2 + p) of those were factorable. Again, don't know if that's related at all but thought I'd throw it out there. Any ideas or theorems would be super helpful, thanks!
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  2. #2
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    Quote Originally Posted by kimberu View Post
    Hi again!
    My problem is to show that a field of order p^2 exists for every prime p.

    In an earlier problem I found that there were p^2monic quadratics in  Z_p[x], but I don't know if that's useful. I also showed that (1/2)(p^2 + p) of those were factorable. Again, don't know if that's related at all but thought I'd throw it out there. Any ideas or theorems would be super helpful, thanks!

    Ok, let p(x)\in\mathbb{F}_p[x]:=\mathbb{Z}_p[x] be one of those monic quadratics which is also irreducible ( can you prove such a thing must always exist? Note that according to what

    you said you can!)) , so that the ideal I:=<p(x)> is maximal in \mathbb{F}_p[x] (why?) \iff the factor ring \mathbb{F}_{p^2}:=\mathbb{F}_p[x]/I is a field (why?) .

    Well, now just prove that \left|\mathbb{F}_{p^2}\right|=p^2 ...

    Hint: every element in \mathbb{F}_{p^2} as defined above has the form f(x)+I\,,\,\,f(x)\in\mathbb{F}_p[x] , but using Euclides algorithm in \mathbb{F}_p[x] show that any element in \mathbb{F}_{p^2} has a

    representative f(x)+I with \deg f<2 and any two of them is a different element in the field.

    Well, now just count up all the different constant and linear poilynomials in \mathbb{F}_p[x] ...

    Tonio
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