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THE P-ADICS, HENSEL’S LEMMA, AND NEWTON POLYGONS

This paper aims to develop the theory of p-adic integers and their basic algebraic and topological properties. It introduces the field of p-adic numbers and the p-adic completion of the rationals, though this is not the main focus of the paper and some proofs are omitted in the interest of brevity. Hensel’s lemma concerning roots of polynomials over Zp is introduced along with limits, concluding with a discussion of Newton polygons. This paper assumes a basic knowledge of analysis, abstract algebra, and Galois theory, though many of the most important definitions are nonetheless provided.