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.