restrictions to the element $b$, such that $b=\sigma (a)$ where $\sigma$ is a generator of the Galois group

Question

Let $K/L$ be a finite Galois extension of degree $p$, $p$ is prime ($char \: L =p$, $p$ is prime). As I know $K=L(a)$ for some $a\in L$. Is there a way to find any restrictions to the element $b$, such that $b=\sigma (a)$ where $\sigma$ is a generator of the Galois group or $b$ can be an arbitrary element? (I know that all these $b$ are roots of the minimal polynomial of $a$, but I’m in the situation that I have no information about the minimal polynomial of $a$. All I know is that the extension is galois and it’s degree is $p$ and $char \: L =p$). Thank you!

Answer 1

The Artin-Schreier theory provides a canonical construction of Galois (cyclic) extensions $L/K$ of degree $p$ of a field $K$ of characteristic $p\neq 0$. Define the A.-S. operator $P$ by $P(x)=x^p -x$. For $\alpha\in K$, but $\notin P(K)$, the A.-S. polynomial $X^p-X-\alpha$ is irreducible in $K[X]$, and its splitting field over $K$ is cyclic of degree $p$, the roots of the A.-S. polynomial being of the form $a,a+1, ..., a+p-1$ (easy). Conversely, any cyclic extension of degree $p$ of $K$ is obtained in this manner (not elementary; this is an additive analog of Kummer theory in characteristic $0$, see e.g. S. Lang's "Algebra", chap.6, §6).