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I'm reading through some notes online concerning finite fields, and attempting to come up with a proof that all finite fields of the same size are isomorphic. But I'm getting stuck at a certain point, and I was wondering of you might have any hints.

If $F$ is any field with $p^d$ elements and $m(x)$ has coefficients all from $\mathbb{F}_p$ and is an irreducible polynomial of degree $d$ over $\mathbb{F}_p$, then $m(x)$ has roots in $F$. Is this immediately obvious? It seems to be stated so but I can't see why.

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    The situation is even slightly better; two finite fields of the same cardinal are in fact equal inside a fixed algebraic closure.2012-07-26
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    Hint: Set $q = p^{d}.$ Then every element $f \in F$ satisfies $f^{q} = f$ (consider non-zero $f$ and use a little elementary group theory).2012-07-26

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