# What is Hensel's lemma a lemma for?

Was Hensel's lemma originally used for proving some other theorem? Or is it meant to be a standalone result? Why is it a "lemma" and not a theorem?

• Look at Hensel's book (1908), which is available online at archive.org/details/theoriederalgeb01hensgoog. This is not the first place where Hensel published one of the various versions of his lemma, but it would be a good place to start since this was the first widely read treatment by him of p-adic numbers. According to section 2.4 of people.math.sfu.ca/~tarchi/SKieffer.pdf Hensel's lemma is treated in chapter 4. – KCd Sep 5 '16 at 14:46
• Of course also keep in mind the term "lemma" in named results (Zorn's lemma, Nakayama's lemma) does not necessarily mean it was logically ever a lemma for some specific theorem. Consider "Bertrand's postulate." – KCd Sep 5 '16 at 14:48
• I looked in Chapter 4 of Hensel's book and he does not separate out results from the main text in the form of separately numbered results (Theorem 4.1, Corollary 4.2, and so on). Section 4.1 is about a proof of a lemma (Hilfssatz), but that lemma is purely algebraic and not what you would call "Hensel's Lemma." See p. 68 for Hensel's Lemma as a statement about lifting polynomial factorizations and p. 71 for Hensel's Lemma as a statement about lifting a root (linear factor). He puts "$(p)$" after an equation to mean it takes places in the $p$-adics (no notation like $\mathbf Q_p$ is used). – KCd Sep 5 '16 at 15:12