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## P-adic number
For every prime number and over the p-adic numbers for every prime p. The space Q_{p} of all p-adic numbers has the nice topological property of completeness, which allows the development of p-adic analysis akin to real analysis.
## Motivation
If _{i} are integers in {0,...,p-1}.
For example, the 2-adic or binary expansion of 35 is 1·2^{5} + 0·2^{4} + 0·2^{3} + 0·2^{2} + 1·2^{1} + 1·2^{0}, often written in the shorthand notation 100011_{2}.The familiar approach to generalizing this description to the larger domain of the rationals (and, ultimately, the real numbers) is to include sums of the form: _{5}. In this formulation, the integers are precisely those numbers which can be represented in the form where a_{i} = 0 for all i<0.
As an alternative, if we extend the k is some (not necessarily positive) integer, we obtain the field Q_{p} of . Those p-adic numbersp-adic numbers for which a_{i} = 0 for all i<0 are also called the .p-adic integers
Intuitively, as opposed to ## Constructions## Analytic approachThe real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers; this allows us to, for example, write 1 as 1.000... = 0.999... . However, the definition of a Cauchy sequence relies on the metric chosen and, by choosing a different one, numbers other than the real numbers can be constructed. The usual metric which yields the real numbers is called the Euclidean metric.
For a given prime
For example with x|_{p} has the effect that high powers of p become "small".
It can be proved that each norm on Q_{p} of p-adic numbers can then be defined as the completion of the metric space (Q,d_{p}); its elements are equivalence classes of Cauchy sequences, where two sequences are called equivalent if their difference converges to zero. In this way, we obtain a complete metric space which is also a field and contains Q.
It can be shown that in k is some integer and each a_{i} is in {0,...,p-1}. This series converges to x with respect to the metric d_{p}.## Algebraic approach
We start with the inverse limit of the rings
_{n≥1} such that a is in
_{n}Z_{pn}, and if n < m,
a = _{n}a (mod _{m}p). ^{n}
Every natural number p-adic integer. For example, in this case 35 as a 2-adic integer would be written as the sequence {1, 3, 3, 3, 35, 35, 35, ...}.
Note that pointwise addition and multiplication of such sequences is well defined, since addition and multiplication commute with the n, a and _{n}p are relatively prime (their greatest common divisor is a), and so _{1}a and _{n}p are relatively prime. Therefore, each ^{n}a has an inverse mod _{n}p, and the sequence of these inverses, (^{n}b), is the sought inverse of (_{n}a). _{n}
Every such sequence can alternatively be written as a series of the form we considered above. For instance, in the 3-adics, the sequence (2, 8, 8, 35, 35, ...) can be written as 2 + 2*3 + 0*3
The ring of n and a p-adic integer u.## Properties
The set of
The
The topology of the set of
The real numbers have only a single proper algebraic extension, the complex numbers;
in other words, this quadratic extension is already algebraically closed.
By contrast, the algebraic closure of the
The number p-adic number for all p except 2, for which one must take at least the fourth power. Thus e is a member of all algebraic extensions of p-adic numbers.
Over the reals, the only functions whose derivative is zero are the constant functions. This is not true over
Given any elements ## Generalizations and related concepts
The reals and the
Suppose NP denotes the (finite) cardinality of D/P. Completing with respect to this norm |.|_{P} then yields a field E_{P}, the proper generalization of the field of p-adic numbers to this setting.Often, one needs to simultaneously keep track of all the above mentioned completions, which are seen as encoding "local" information. This is accomplished by adele rings and idele groups. | |||||||

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