finite field (and subfields thereof)

**mylestone** · April 23rd 2009, 12:12 AM

$K$ is a finite field, $F$ is a subfield of $K$ , and $m$ is a positive integer.

$L=\{a \in K \vert a^{p^m} \in F \}$ .

Show that $L$ is a subfield of $K$ containing $F$ . Moreover, show that $L=F$ .

Easy enough to show $L$ is a subfield of $K$ , and I have a proof that $L=F$ assuming that $L$ contains $F$ , but I do not know why $L$ must contain $F$ .

Thanks in advance (and afterwards as well, naturally).

**NonCommAlg** · April 23rd 2009, 05:23 PM

Originally Posted by mylestone

$K$ is a finite field, $F$ is a subfield of $K$ , and $m$ is a positive integer.

$L=\{a \in K \vert a^{p^m} \in F \}$ .

Show that $L$ is a subfield of $K$ containing $F$ . Moreover, show that $L=F$ .

Easy enough to show $L$ is a subfield of $K$ , and I have a proof that $L=F$ assuming that $L$ contains $F$ , but I do not know why $L$ must contain $F$ .

you're kidding, right? well, if $a \in F \subseteq K,$ then obviously $a^{p^m}$ will also be in $F$ (because F is closed under multiplication). thus $F \subseteq L.$

**mylestone** · April 23rd 2009, 11:54 PM

Originally Posted by NonCommAlg

you're kidding, right? well, if $a \in F \subseteq K,$ then obviously $a^{p^m}$ will also be in $F$ (because F is closed under multiplication). thus $F \subseteq L.$

Jeepers ...apologies, and thank you.

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