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I know thar $\forall p$ prime, $\forall n>0$, it exists the finite field $GF(p^n)$. Can you help me proving this theorem? I do not need a formal proof, just an intuition, an idea...

Thank you

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    hint:you can use polynomial $x^{p^n}-x\in Z_p[X] $2012-01-11

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Hint: prove that the set of all $x$ such that $x^{p^n}-x=0$ is a field.

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HINT:

First, constructing a finite field with $p = p^1$ elements is easy.

For $n > 1$: Consider the ring $R = (GF(p))[x]/E(x)$ where $E(x)$ is degree $n$ polynomial (you may even assume $E$ is monic). How many elements does $R$ have? Under what conditions on $E(x)$ is $R$ a field?

Note: As Steven points out, showing the existence of a polynomial $E(x)$ with the required properties is quite nontrivial. I am just hoping this is a fruitful direction for you to think about.

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    @DilipSarwate, but how to prove it?2012-01-11