p -adic number

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For every prime number , the p -adic numbers form an expansion field of the field of rational numbers ; they were first described by Kurt Hensel in 1897 . These fields are used to solve problems in number theory , often using the local-global principle of Helmut Hasse , which - to put it simply - says that an equation can be solved over the rational numbers if and only if it is over the real numbers and can be solved over all (which is not so generally true, for the exact meaning see there). As a metric space is complete and thus allows the development of a -adic analysis analogous to real analysis .


If a fixed prime number, then every integer can be in an -adic expansion of the form

(it is said that the number is written down as the base , see also the place value system ), with the digits off . For example, the 2-adic expansion is precisely the binary representation; for example one writes:

The well-known generalization of this description to larger sets of numbers (rational and real) is the generalization to infinite sums at the lower end, i.e. H. of the following form:

These series are convergent with respect to the ordinary absolute value. For example, the 5-adic representation is from to the base . In this system, the integers are exactly those that hold true for everyone .

But one can also define a concept of convergence in which the sums at the other end are extended to infinity, and so rows of form


where is any integer. In this way we get the field of -adic numbers (as opposed to the (real) numbers that are represented in an (ordinary) -adic place value system). Those -adic numbers that hold for all are called whole -adic numbers. Analogous to the usual -adic expansion , one can write this series as a sequence of digits (infinitely continued to the left):

The convention of putting the ellipses on the left side reflects the reading direction, but has the advantage that finite symbol sequences, which have the same meaning in both cases, do not differ in the notation.

The usual -adic expansion thus consists of sums that continue to the right with ever smaller (negative) powers of , and the -adic numbers have developments that continue to the left with ever larger -potencies.

With these formal Laurent series in one can calculate like with the usual -adic expansions of real numbers: addition from right to left with carryover, multiplication according to the school method. You just have to note that carryovers can continue into infinity, for example adding and gives the number . A sign is not needed because all additive inversesthere are no negative numbers  - have an -adic representation (1) .

Furthermore, the subtraction can be carried out according to the school method from right to left, possibly with an infinitely often occurring return carryover (try it with ).

In contrast to the school method, the division is also carried out from right to left, so the result is continued to the left if the division does not work.

The question remains whether these series make sense at all; H. whether they converge in any sense. Two solutions for this are now presented.


Analytical construction

The real numbers can be constructed as the completion of the rational numbers, whereby they are understood as equivalence classes of rational Cauchy sequences . This allows us, for example, to write the number as or as , since in is true.

However, the definition of a Cauchy sequence already depends on the metric used , and for a metric other than the usual Euclidean ( Archimedean ) metric that is generated from the absolute value, other completions result instead of the real numbers.

p -adic amount

For a fixed prime the defining p -adischen amount to : Any rational number can be in the form of writing with a uniquely determined integer and two natural numbers and both not divisible. The -adic amount is then defined as

and .

This is a non-Archimedean amount .

For example :

for any other prime number

In the sense of this amount , large powers are small in terms of amount . This defines a discrete evaluation ring on the -adic numbers .

Exponent evaluation

It is often convenient (and common practice in the literature) to introduce a different notation for non-Archimedean evaluations. Instead of the absolute value , one chooses the exponent . The definition relations of the evaluation are as follows in the exponents:

  1. for .
  2. .
  3. .
  4. .

One speaks of an exponent weighting , sometimes also p- weighting , and of an exponentially weighted ring or body. The transition to the exponents is made possible by the fact that, due to the tightened triangle inequality, the values do not need to be added. The formation of the logarithms reverses the order and transforms the multiplication into an addition.

Often one normalizes so that is for the prime element .

p -adic metric

The p -adic metric on is defined by the amount:

Thus, for example, the sequence in is a zero sequence with regard to the 5-adic metric, whereas the sequence is restricted, but not a Cauchy sequence, because the following applies to each :

The completion of the metric space is the metric space of the -adic numbers. It consists of equivalence classes of Cauchy sequences, with two Cauchy sequences being equivalent if the sequence of their pointwise -adic distances is a null sequence . In this way one obtains a complete metric space, which is also contained (through the well-defined component-wise linkages of the Cauchy sequence equivalence classes) .

Since the metric defined in this way is an ultrametric , series already converge when the summands form a zero sequence. In this body, then, are the series of forms mentioned above

instantly recognizable as a convergent, if an integer and in lying. One can show that each element of exactly one such series can be represented as a limit value .

Algebraic construction

Here the ring of whole -adic numbers is defined first, and then its quotient field .

We define a projective Limes

of the remainder class rings : An integer -adic number is a sequence of remainder classes that meet the compatibility condition (of the projective limit)

fulfill. For every integer the (stationary) sequence is an element of . Is in this way in embedded , then is sealed in .

The addition and multiplication, which are defined by components, are well-defined, since the addition and multiplication of whole numbers can be interchanged with the formation of the remainder of the class. So every -adic integer has the additive inverse ; and every number whose first component is   not   has a multiplicative inverse, because in this case all are too coprime, i.e. have an inverse modulo , and the consequence (which also fulfills the compatibility condition of the projective limit) is then the inverse to .

Each -adic number can also be represented as a series of the form (1) described above , then the components of the series are calculated using the partial sums

educated. For example, the -adic sequence can also be used as

or in the shortened form as .

The ring of whole -adic numbers does not have zero divisors , so we can form the quotient field and get the field of -adic numbers. Each of the different elements of this body can be represented in the form , where is an integer and a unit in . This representation is clear.

Furthermore applies


The amount of units is often using

and the set of units with

Both are multiplicative groups and it applies

with as the sign for the finite body with elements and as the sign for the direct product .


  • The set of -adic integers (and the set of -adic numbers) is uncountable . This means that there are non-rational and non-algebraic, i.e. transcendent numbers in .
  • is a whole body .
  • The field of -adic numbers contains and therefore has characteristics , but cannot be arranged . 
  • The topological space of the whole -adic numbers is a totally disconnected compact space , the space of all -adic numbers is locally compact and totally disconnected. As metric spaces, both are complete .
  • The prime elements of are exactly the elements associated with the number . These are exactly those elements whose amount is equal ; this amount is the largest amount occurring in that is less than . The prime elements of finite extensions of are divisors of .
  • is a local ring , more precisely a discrete evaluation ring . Its maximal ideal is generated by (or any other prime element).
  • The remainder class field of is the finite field with elements.
  • (and ) contains the -th roots of unity (see Hensel's lemma ). For these are all roots of unity; their group is isomorphic to Für and the root of unity is added.
  • Is a primitive -th root of unity in then is a monoid and as an alternative to the numeric system in (1) the system used for each there and with and
All results are clear, is the same as in (1) . is called the system of Teichmüller representatives .
  • The metric on can be continued to a metric on the algebraic closure, but this is then not complete. The completion of the algebraic conclusion with regard to this metric leads to the field which corresponds approximately to the complex numbers with regard to its analysis.

p -adic function theory

The power series

the exponential function has its coefficient in . It converges for all with . This radius of convergence applies to all algebraic extensions of and their completions, including

That is in for everyone ; in lies . There are algebraic expansions of in which the -th root of or the fourth root of lies; these roots could be understood as -adic equivalents of Euler's number . However, these numbers have little to do with Euler's real number .

The power series

for the logarithm converges for .

The following applies in the convergence areas



The functional equations known from real and complex analysis also apply there .

Functions from to with derivative are constant. This theorem does not apply to functions from to ; for example has the function

for ,

is entirely the derivative , but is not even locally constant in . The derivative is defined analogously to the real case via the limit value of the difference quotient, and the derivative is in


Differences to the Archimedean systems

Apart from the other convergence of the -adic metric compared to the Archimedean metric described under the place value system , there are other (resulting) differences:

  • The -adic bases are prime numbers or prime elements . There are zero divisors for composite whole numbers as bases. In order not to awaken the association of a body, the spelling is avoided and used instead . Nonetheless, a ring is, if not an area of integrity . Are two different prime numbers, then though . With real numbers, however, any whole number can play the role of the base, and algorithms exist to convert the representation of a real number in one base into that of another.
  • The (canonical) -adic representation of a number in as an infinite sum (1) is unique. On the other hand, there are fractions for every base of a place value system for real numbers , for which there are two representations as an infinite sum, as in the case of decimal or balanced ternary .


  • The representation of in canonical format (1) is .
  • Since the number in can be represented as a sum of squares for all prime numbers , it can not be arranged .As a result, there is no sign as with real numbers (as a prefix to a number constant), not even “negative” numbers.
  • The algorithms z. B. for the basic arithmetic operations (addition, subtraction, multiplication and division) all run from finite (right) to potentially infinite (left). Carries work in the same (ascending) direction to the left neighbor. If the calculation is canceled, you can immediately indicate the size of the error. With the division algorithm, the next digit of the quotient can be obtained by multiplying.
    With addition, subtraction and multiplication in the real place value systems, one can also start with the lower powers in the case of interrupting representations and work in the carryovers progressively to higher powers.
    However, if you want to start in the finite (e.g. with irrational numbers ) (on the left with the high powers) and progress to small powers (i.e. too great accuracy), then the carryovers work in the opposite direction and it is an error estimate for ensuring the correctness of the digit to be printed required.
  • A non-Archimedean metric defines an equivalence relation for each .
  • For and one receives such an evaluation ring as one that contains at least one, or , forever , but does not represent the whole body. There is nothing comparable with the Archimedean systems.
  • Topologically, they are compact and totally incoherent , while those locally are compact and totally incoherent. is locally compact and simply connected .

Approximation theorem

If there are elements of , then there is a sequence in such that for each (inclusive ) the limit of in is under . (This statement is sometimes called an approximation theorem.)

See also


Web links

References and comments

  1. a b There are authors who, in the case of periodic representations, place the base directly next to the comma on the side on which the series continues into infinity, i.e.: and or .
  2. Convergence can only take place on one of the two sides, so that the development on at least one side must be finite.
  3. Since every power of divides the 0, is as usual for everyone .
  4. ^ Van der Waerden : Algebra , Part Two . Springer-Verlag, 1967, Evaluated Bodies, p. 204 f .
  5. Normalized in this way, the exponent evaluation corresponds to the order of a formal power series in with the indefinite as the prime element.
  6. Leutbecher, 1996 , p. 118 f.
  7. Leutbecher, 1996 , p. 117 f.
  8. a b c Baker, 2007 , Theorem 4.33.
  9. An example is given in Pro-Ending Number # 10-adic Numbers .
  10. Furthermore, irrational algebraic numbers can look completely different in the fields. For example, is , however , and the 2-adic development , whereas the 5-adic is. The two developments can (as always) be combined with the Chinese remainder of the sentence . Other prime numbers with are . (See also Gérard P. Michon: Solving algebraic equations ).


  11. http://www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf p.26
  12. ^ Gérard P. Michon: Dividing two p-adic numbers