# 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
{"ft_lang_label":"__label__en","ft_lang_prob":0.91687185,"math_prob":0.9795104,"size":1894,"snap":"2019-43-2019-47","text_gpt3_token_len":524,"char_repetition_ratio":0.12592593,"word_repetition_ratio":0.046728972,"special_character_ratio":0.26187962,"punctuation_ratio":0.097297296,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9907174,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-20T19:14:50Z\",\"WARC-Record-ID\":\"<urn:uuid:99b24d9e-df24-44ab-b667-1eb4aad8d395>\",\"Content-Length\":\"135782\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2a865dc4-def2-437f-8eb3-4074a648d262>\",\"WARC-Concurrent-To\":\"<urn:uuid:23d82b93-ad38-4625-8579-a44a41b89e29>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/3239252/why-p-cyclic-extension-iff-pm-cyclic-extension-forall-m?noredirect=1\",\"WARC-Payload-Digest\":\"sha1:TW4E4IRCN3Q577VYKUI55OIZP2OGWQGB\",\"WARC-Block-Digest\":\"sha1:OLRQBIF5ET76AVRTM3NQHQYPSZDXYC7C\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496670601.75_warc_CC-MAIN-20191120185646-20191120213646-00016.warc.gz\"}"}