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Let $E$ and $F$ be subfields of $GF(p^n)$. If $|E| = p^r$ and $|F| = p^s$, what is the order of $E\cap F$?

I read a corollary that "A finite field of order $p^n$ contains a unique subfield of order p^m for each $m$ | $n$ and no other subfields.

If that's the case, wouldn't that mean if two subfields have different orders, their intersection is $0$? So in this case, the order of $E \cap F$ = $0$?

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    $\begingroup$ The intersection can't be zero since it contains the prime subfield $GF(p)$! And anyway, $1 \in E, F$! $\endgroup$ – Robert Lewis Apr 6 '14 at 6:04
  • $\begingroup$ Very good point, thank you for the insight. $\endgroup$ – James Apr 6 '14 at 6:33
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    $\begingroup$ If you think about that corollary, you should be able to answer your question. $\endgroup$ – Gerry Myerson Apr 6 '14 at 6:35
  • $\begingroup$ Any source of this corollary in any text or notes will be helpful!. $\endgroup$ – BAYMAX Dec 23 '17 at 3:25
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Extended hint:

So if $t\mid r$, and $t\mid s$, then, by the result you quoted, both $E$ and $F$ contain a unique copy of $GF(p^t)$. But also $GF(p^n)$ contains a unique copy of $GF(p^t)$, which means that the copies of $GF(p^t)$ inside $E$ (resp. $F$) must coincide with the one in $GF(p^n)$, and thus be contained in the intersection $E\cap F$.

In view of this the intersection $E\cap F$ is a copy of $GF(p^\ell)$, where $\ell=\gcd(r,s)$.

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  • $\begingroup$ So if $E$ and $F$ contain a unique copy of $GF(p^t)$ and $GF(p^n)$ also contains a unique copy of $GF(p^t)$ then $|E \cap F|$ $\geq$ $p^t$, and the intersection also contains the prime subfield $GF(p)$ as well. So... $|E \cap F| = p^t + p$ in this case? $\endgroup$ – James Apr 6 '14 at 6:43
  • $\begingroup$ @James: The prime field is contained in all the copies of $GF(p^t)$. You should also realize that the intersection has to be a subfield. The question is: which subfield? Or what is the biggest common subfield? $\endgroup$ – Jyrki Lahtonen Apr 6 '14 at 6:51
  • $\begingroup$ so with that. Using your example $GF(p^t)$, wouldn't the biggest subfield ($E \cap F$) just be the biggest $t$ that divides $r$, $s$ and $n$? $\endgroup$ – James Apr 6 '14 at 7:08
  • $\begingroup$ Correct, @James! $\endgroup$ – Jyrki Lahtonen Apr 6 '14 at 7:41
  • $\begingroup$ Thank you for the hints, helped me out. :) $\endgroup$ – James Apr 6 '14 at 15:07

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