New light on Hensel's lemma |
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Authors: | David Brink |
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Institution: | aDepartment of Mathematics, Universitetsparken 5, 2100 Copenhagen, Denmark |
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Abstract: | The historical development of Hensel's lemma is briefly discussed (Section 1). Using Newton polygons, a simple proof of a general Hensel's 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 Krasner's lemma (Section 7), we give a short proof of a general Hensel's 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. |
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Keywords: | Valued fields Hensel's lemma Krasner's lemma Newton polygons Continuity of roots |
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