Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$. I am interested in determining the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$ if $K$ is an infinite field. In the case where $K$ is a finite field, it is commonly assumed, without loss of generality, that $\sigma$ is the Frobenius map: $x \mapsto x^q$, where $|K| = q$. Similarly, in the case of an infinite field with positive characteristic, can we give a suitable generator $\sigma$ for the Galois group $\text{Gal}(L/K)$? Especially when $ K $ is a function-field.
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If $K$ is finite with order $q$, then we have the nice theorem that the $q$th power map generates ${\rm Gal}(L/K)$ when $L$ is an arbitrary finite extension of $K$, but there is no analogue of this when $K$ is infinite.
In the case of a "named" extension (Kummer, Artin-Schrier, etc.) you may have a nice description of a generator of the Galois group, but in general there's not going to be an explicit description of a generator. When $K$ is a function field over a finite field, look up class field theory in characteristic $p$, which is... complicated.
