New light on Henselʹs lemma

The historical development of Henselʹs lemma is briefly discussed (Section 1). Using Newton polygons, a simple proof of a general Henselʹs lemma for separable polynomials over Henselian fields is given (Section 3). For polynomials over algebraically closed, valued fields, best possible results on continuity of roots (Section 4) and continuity of factors (Section 6) are demonstrated. Using this and a general Krasnerʹs lemma (Section 7), we give a short proof of a general Henselʹs lemma and show that it is, in a certain sense, best possible (Section 8). All valuations here are non-Archimedean and of arbitrary rank. The article is practically self-contained.