Let $L / K / M$ be field extensions. Define the degree $[K: M]$ of the field extension $K / M$, and state and prove the tower law.

Now let $K$ be a finite field. Show $\# K=p^{n}$, for some prime $p$ and positive integer $n$. Show also that $K$ contains a subfield of order $p^{m}$ if and only if $m \mid n$.

If $f \in K[x]$ is an irreducible polynomial of degree $d$ over the finite field $K$, determine its Galois group.