This is exercise $6.29$ out of Lang's book:
Let $K$ be a cyclic extension of a field $F$, with Galois group $G$ generated by $\sigma$. Assume that the characteristic is $p$, and that $[K:F]=p^{m-1}$ for some $m\geq2$. Let $\beta$ be an element of $K$ such that that Tr$^K_F(\beta)=1$.
(a) Show that there exists an element $\alpha\in K$ such that $\sigma(\alpha)-\alpha=\beta^p-\beta$.
(b) Prove that the polynomial $x^p-x-\alpha$ is irreducible in $K[x]$.
(c) If $\theta$ is a root of this polynomial, prove that $F(\theta)$ is a Galois, cyclic extension of degree $p^m$ of $F$, and that its Galois group is generated by an extension $\sigma^*$ of $\sigma$ such that $\sigma^*(\theta)=\theta+\beta$.
I have been able to do letter $a$ using Hilbert's Theorem $90$ (Additive Form), since $$\text{Tr}(\beta)=1=1^p=(\text{Tr}(\beta))^p=\text{Tr}(\beta^p).$$ I'm at a loss for the second one, even though it seems to scream the Artin-Schreier theorem. For the third part, I certainly see that it is an extension of degree $p^m$, although I'm not sure I can get much farther than that. How can I do the last two parts?
Best Answer
For (b), suppose that $\gamma\in K$ satisfies $\gamma^p-\gamma=\alpha$, and make a construction that relates $\gamma$ to $\beta$. Once you've related $\gamma$ to $\beta$, you can get a contradiction from the fact that $\beta$ has trace $1$.
For (c), first show that there is an extension of $\sigma$ to an automorphism $\sigma^*$ of $K(\theta)$ such that $\sigma^*(\theta)=\theta+\beta$. Then show that $(\sigma^*)^i$ fixes $\theta$ if and only if $p^m\mid i$. Then deduce that $K(\theta)=F(\theta)$.