How can I prove that any field of order $p^n$ has a unique subring of order $p^m$ where $m$ divides $n$. I understand that there is a unique subfield of order $p^m$ for every $m$. But I can not see that there is no other subring of order $p^m$.
Can anyone please explain me?
Every subring of a finite field is a field. If $a$ is a nonzero element of a finite field $F$, then $a^n=1$ for some $n>0$, so $a^{n-1}=a^{-1}$. Any subring of $F$ containing $a$ therefore contains $a^{-1}$.