(a) State and prove the tower law.

(b) Let $K$ be a field and let $f(x) \in K[x]$.

(i) Define what it means for an extension $L / K$ to be a splitting field for $f$.

(ii) Suppose $f$ is irreducible in $K[x]$, and char $K=0$. Let $M / K$ be an extension of fields. Show that the roots of $f$ in $M$ are distinct.

(iii) Let $h(x)=x^{q^{n}}-x \in K[x]$, where $K=F_{q}$ is the finite field with $q$ elements. Let $L$ be a splitting field for $h$. Show that the roots of $h$ in $L$ are distinct. Show that $[L: K]=n$. Show that if $f(x) \in K[x]$ is irreducible, and deg $f=n$, then $f$ divides $x^{q^{n}}-x$.

(iv) For each prime $p$, give an example of a field $K$, and a polynomial $f(x) \in K[x]$ of degree $p$, so that $f$ has at most one root in any extension $L$ of $K$, with multiplicity $p$.