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For every prime number p, the padic numbers form an extension field of the rational numbers first described by Kurt Hensel[?] in 1897. They have been used to solve several problems in number theory, many of them using Helmut Hasse[?]'s localglobal principle[?], which roughly states that an equation can be solved over the rational numbers if and only if it can be solved over the real numbers and over the padic numbers for every prime p. The space Q_{p} of all padic numbers has the nice topological property of completeness, which allows the development of padic analysis[?] akin to real analysis.

If p is a fixed prime number, then any integer can be written as a padic expansion (usually referred to as writing the number in "base p") in the form
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:
As an alternative, if we extend the padic expansions by allowing infinite sums of the form
Intuitively, as opposed to padic expansions which extend to the right as sums of ever smaller, increasingly negative powers of the base p (as is done for the real numbers as described above), these are numbers whose padic expansion to the left are allowed to go on forever. The main technical problem is to define a proper notion of infinite sum which makes these expressions meaningful; two different but equivalent solutions to this problem will be presented below.
The 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 p, we define the padic metric in Q as follows: for any nonzero rational number x, there is a unique integer n allowing us to write x = p^{n}(a/b), where neither of the integers a and b is divisible by p. Unless the numerator or denominator of x contains a factor of p, n will be 0. Now define x_{p} = p^{n}. We also define 0_{p} = 0.
For example with x = 63/550 = 2^{1} 3^{2} 5^{2} 7 11^{1}
This definition of x_{p} has the effect that high powers of p become "small".
It can be proved that each norm on Q is equivalent either to the Euclidean norm or to one of the padic norms for some prime p. The padic norm defines a metric d_{p} on Q by setting
It can be shown that in Q_{p}, every element x may be written in a unique way as
In the algebraic approach, we first define the ring of padic integers, and then construct the field of quotients of this ring to get the field of padic numbers.
We start with the inverse limit of the rings Z_{pn} (see modular arithmetic): a padic integer is then a sequence (a_{n})_{n≥1} such that a_{n} is in Z_{pn}, and if n < m, a_{n} = a_{m} (mod p^{n}).
Every natural number m defines such a sequence (m mod p^{n}), and can therefore be regarded as a padic integer. For example, in this case 35 as a 2adic 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 mod operator, see modular arithmetic. Also, every sequence (a_{n}) where the first element is not 0 has an inverse: since in that case, for every n, a_{n} and p are relatively prime (their greatest common divisor is a_{1}), and so a_{n} and p^{n} are relatively prime. Therefore, each a_{n} has an inverse mod p^{n}, and the sequence of these inverses, (b_{n}), is the sought inverse of (a_{n}).
Every such sequence can alternatively be written as a series of the form we considered above. For instance, in the 3adics, the sequence (2, 8, 8, 35, 35, ...) can be written as 2 + 2*3 + 0*3^{2} + 1*3^{3} + 0*3^{4} + ... The partial sums of this latter series are the elements of the given series.
The ring of padic integers has no zero divisors, so we can take the quotient field to get the field Q_{p} of padic numbers. Note that in this quotient field, every number can be uniquely written as p^{n}u with a natural number n and a padic integer u.
The set of padic integers is uncountable.
The padic numbers contain the rational numbers Q and form a field of characteristic 0. This field cannot be turned into an ordered field.
The topology of the set of padic integers is that of a Cantor set; the topology of the set of padic numbers is that of a Cantor set minus a point (which would naturally be called infinity). In particular, the space of padic integers is compact while the space of padic numbers is not; it is only locally compact. As metric spaces, both the padic integers and the padic numbers are complete.
The real numbers have only a single proper algebraic extension, the complex numbers; in other words, a quadratic extension is already algebraically closed. By contrast, the algebraic closure of the padic numbers has infinite degree. Furthermore, Q_{p} has infinitely many inequivalent algebraic extensions.
The number e, defined as the sum of reciprocals of factorials, is not a member of any padic field; but e^{p} is a padic 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 padic numbers.
Over the reals, the only functions whose derivative is zero are the constant functions. This is not true over Q_{p}. For instance, the function f(x) = (1/x_{p})^{2} has zero derivative.
Given any elements r_{∞}, r_{2}, r_{3}, r_{5}, r_{7}, ... where r_{p} is in Q_{p} (and Q_{∞} stands for R), it is possible to find a sequence (x_{n}) in Q such that for all p (including ∞), the limit of x_{n} in Q_{p} is r_{p}.
The reals and the padic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general algebraic number fields, in an analogous way. This will be described now.
Suppose D is a Dedekind domain and E is its quotient field. The nonzero prime ideals of D are also called finite places or finite primes of E. If x is a nonzero element of E, then xD is a fractional ideal and can be uniquely factored as a product of positive and negative powers of finite primes of E. If P is such a finite prime, we write ord_{P}(x) for the exponent of P in this factorization, and define
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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