In Guillot’s “A gentle course in Local Class Field Theory” Proposition 1.7 describes the relation between extensions by n-roots and their Galois groups. It is quoted below.
The radical extensions produce cyclic Galois groups and if we know we have a cyclic group, the extension can be constructed via a radical. Guilliot indicates that the two statements are not the convers of each other. By this I understand that while in (2) the cyclic group leads to an extension by an n-th root, in (1) the $n$-th root can lead to a smaller Galois group ($|G|<n$) as the group generated by the element $a$ in the residue class in can be a proper subgroup of those classes. Guillot indicates that the result is more symmetric in the case n=p and offers without proof the following corollary:
I need help understanding the requirement of roots of order $p^2$. I expected this result to use roots of unity of order p instead. Looking ahead in the text the corollary is used and explicitly references the use of the second power so this is not a typo. Can you provide a proof of 1.9 or an informal justification of the $p^2$ condition? What goes wrong when only the roots of order p are used?
If I use just p roots of unity in proposition 7, an element $a$ of $F^\times/F^{\times p}$ can only have order $1$ or $p$. In the first case the extension is trivial and in the second the extension is cyclic of degree $p$. If a cyclic extension of order $p$ exists it would be a radical extension. I have considered the possibility that $p=2$ might be a spoiling case. I have also considered that a field of characteristic $p$ might cause problems. I have not found the issue in those cases.