# Thread: finite field (and subfields thereof)

1. ## finite field (and subfields thereof)

$K$ 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).

2. 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 $F \subseteq L.$

3. 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.