# Why p-cyclic extension iff $p^m$-cyclic extension $\forall m$

A theorem is stated as follows. For a field $$F$$ of characteristic $$p$$, $$F$$ has a $$p$$-cyclic extension if and only if for every positive integer $$m$$, $$F$$ has a $$p^m$$-cyclic extension. I wonder if there is an elementary proof of it, without too much discussion on Abelian extension theories. All I am acquainted with is basic Galois group and group theory.

• You do know that when the extension degree equals the characteristic, Kummer-extensions need to be replaced with Artin-Schreier? This standard argument can be extended to show that $F$ has a cyclic extension of degree $p$ if and only if the mapping $T:F\to F, z\mapsto z^p-z$ is not surjective. Lifting that to cyclic extensions of degree $p^m$ may be a bit more difficult in general. The route I'm aware of would go via Witt vectors, but it sounds like that is one of the pieces you would rather avoid... – Jyrki Lahtonen May 26 at 5:10
• @JyrkiLahtonen I appreciate it. So if I have to prove this as quickly as possible, what’s your advice? Do I need to finish the whole chapter 3 of Jacobson III？ – user11546831 May 26 at 5:16
• Anyway, here is my take on the "easiest" inductive step of going from a cyclic extension of degree two to one of degree four. All using Witt-vector arithmetic. IIRC the arithmetic details become progressively hairier as $p^m$ grows. OTOH you only want existence. May be that is easier, but I am too ignorant. – Jyrki Lahtonen May 26 at 5:18
• @JyrkiLahtonen THX very much – user11546831 May 26 at 5:23
• I'm afraid I don't know if there is an easier way. Jacobson is good. If you already worked your way through the corresponding pieces of Kummer theory, then your background is strong enough. – Jyrki Lahtonen May 26 at 5:24