“p-adic numbers”的概念、定义、翻译、参考文献-科学参考

p-adic numbers

Mathematics

The p-adic numbers were introduced by Kurt Hensel in 1897 with the aim of applying power series methods to number theory. Here p is a prime number.

A p-adic integer is a sequence of the form (x₀, x₁, x₂, …) such that x_n ≡ x_n−1 mod pⁿ for each n. Such sequences arise naturally when investigating congruences such as x² ≡ c mod pⁿ. A more natural representation of a p-adic integer is as

$\sum_{i = 0}^{∞} a_{i} p^{i} where x_{n} = \sum_{i = 0}^{n} a_{i} p^{i}$

and 0≤a_i<p for all i. A p-adic number is then a series of the form

$\sum_{i = k}^{\infty} a_{i} p^{i} with 0 \leq a_{i} < p for all i,$

where k is any integer (possibly negative). The p-adic numbers then naturally form a field denoted ℚ_p.

There is an alternative analytic construction of ℚ_p. Any non-zero rational number x can be uniquely written as pⁿ(a/b), where neither a nor b is divisible by p. The order |x|_p of x is defined to be p^–n, with the order of 0 being 0, and a metric can be defined on ℚ by d(x,y) = |x–y|_p. The p-adic numbers can then be constructed as the completion of ℚ using this metric.

单词	p-adic numbers
释义	p-adic numbers Mathematics The p-adic numbers were introduced by Kurt Hensel in 1897 with the aim of applying power series methods to number theory. Here p is a prime number. A p-adic integer is a sequence of the form (x₀, x₁, x₂, …) such that x_n ≡ x_n−1 mod pⁿ for each n. Such sequences arise naturally when investigating congruences such as x² ≡ c mod pⁿ. A more natural representation of a p-adic integer is as $\sum_{i = 0}^{∞} a_{i} p^{i} where x_{n} = \sum_{i = 0}^{n} a_{i} p^{i}$ and 0≤a_i<p for all i. A p-adic number is then a series of the form $\sum_{i = k}^{\infty} a_{i} p^{i} with 0 \leq a_{i} < p for all i,$ where k is any integer (possibly negative). The p-adic numbers then naturally form a field denoted ℚ_p. There is an alternative analytic construction of ℚ_p. Any non-zero rational number x can be uniquely written as pⁿ(a/b), where neither a nor b is divisible by p. The order \|x\|_p of x is defined to be p^–n, with the order of 0 being 0, and a metric can be defined on ℚ by d(x,y) = \|x–y\|_p. The p-adic numbers can then be constructed as the completion of ℚ using this metric.
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