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Author: Admin | 2025-04-28
Field of rational numbers. This allows considering the p-adic numbers as an extension field of the rational numbers, and the rational numbers as a subfield of the p-adic numbers.The valuation of a nonzero p-adic number x, commonly denoted is the exponent of p in the first nonzero term of every p-adic series that represents x. By convention, that is, the valuation of zero is This valuation is a discrete valuation. The restriction of this valuation to the rational numbers is the p-adic valuation of that is, the exponent v in the factorization of a rational number as with both n and d coprime with p.The p-adic integers are the p-adic numbers with a nonnegative valuation.A -adic integer can be represented as a sequenceof residues mod for each integer , satisfying the compatibility relations for .Every integer is a -adic integer (including zero, since ). The rational numbers of the form with coprime with and are also -adic integers (for the reason that has an inverse mod for every ).The p-adic integers form a commutative ring, denoted or , that has the following properties.The last property provides a definition of the p-adic numbers that is equivalent to the above one: the field of the p-adic numbers is the field of fractions of the completion of the localization of the integers at the prime ideal generated by p.Topological properties[edit]The p-adic valuation allows defining an absolute value on p-adic numbers: the p-adic absolute value of a nonzero p-adic number x iswhere is the p-adic valuation of x. The p-adic absolute value of is This is an absolute value that satisfies the strong triangle inequality since, for every x and y one hasMoreover, if one has This makes the p-adic numbers a metric space, and even an ultrametric space, with the p-adic distance defined byAs a metric space, the p-adic numbers form the completion of the rational numbers equipped with the p-adic absolute value. This provides another way for defining the p-adic numbers. However, the general construction of a completion can be simplified in this case, because the metric is defined by a discrete valuation (in short, one can extract from every Cauchy sequence a subsequence such that the differences between two consecutive terms have strictly decreasing absolute values; such a subsequence is the sequence of the partial sums of a p-adic series, and thus a unique normalized p-adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized p-adic series instead of equivalence classes of Cauchy sequences).As the metric is defined from a discrete valuation, every open ball is also closed. More precisely, the open ball equals the closed ball where v
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