Pro-finite number

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In algebra and number theory , a per-finite number (also per-finite number , per-finite integer or profinite (whole) number , English: profinite integer ) is defined by the remainders (residue classes) that it forms in all integer residue class rings . This makes it an element of the pro-finite completion (spoken: Zett-Dach) of the group of integers . The (rational) integers can be determined by means of the canonical injective homomorphism

embed in the per-finite numbers. The number in all residual class rings is mapped to the one there . This "creates" to a certain extent

The whole numbers embedded in this way lie close to the pro-finite whole numbers. They are in sequences of remainder classes , and with the properties of such a 1 or 2, for example (as is often the case in abstract algebra ), only those are important that they are in their combinations to have.

The Galois group of the algebraic closure of a finite field over this field is isomorphic to


The perfinite completion of the group of integers is

          (projective or inverse Limes ).

The formation of a projective limit requires a so-called projective system consisting of a directed index set , which indexes a sequence of objects, and transition morphisms between these objects. For one takes as directed index set the natural numbers which are partially ordered by the divisibility relation , and as a sequence of objects the sequence of finite cyclic groups For each and every one there is the group homomorphism (the "remainder class mapping", the "natural surjection")

which is well-defined because of . These homomorphisms are taken as transition morphisms between the objects. They map a producer of into one of and are for in one way, viz


tolerable , as is required for the projective system and the formation of the projective limit.

In projective limit those families are grouped by residual classes whose components are compatible with each other, so having all of the following applies:

what by the congruences

is fulfilled. Written in a formula it results:

A family of elements that meet the compatibility requirements, which is part of the projective Limes, is sometimes also referred to as "fiber".

The addition, defined by component, is continuous. The same also applies to multiplication. This becomes a topological additive group and a topological ring with 1.

The natural topology on is the Limes topology , i. i. from the discrete topologies to the induced product topology. This topology is compatible with the ring operations and is also called the Krull topology . At the same time, the closed shell of is in the product, which implies the tightness of in .

Alternative construction

The ring of integers can one also in "classic" style uniform structure completed be. Be for this

a neighborhood (of order ). The set is a (countable) fundamental system and the associated filter

a uniform structure for . The demands on are easily verified:

(1) Each neighborhood and each contains the diagonal
(2) Is and , then is
(3) Is , then is also
(4) For each there is a with .
(5) Is , then is also

The Amount of Cauchy nets in is

which is a group with the component-wise addition. The completion of the whole numbers with regard to the uniform structure of the divisibility is the factor group of the Cauchy networks modulo the zero sequences (more precisely: the sequences that are zero networks or Cauchy networks with limes ).

turns out to be isomorphic to

Be a family of compatible residual classes, so

and be , then is for everyone with


the consequence of the representatives is a Cauchy network.

Conversely, if a sequence of integers is a Cauchy net in the sense of the above-defined uniform structure, then, for every one , so for all with valid


If you take it now , then it is

for everyone with . The subsequence has the same limit as the original sequence, so it represents the same element . Is well , then for everyone with also and , well


The sequence of residual classes thus fulfills the compatibility requirements and is a family

One can move from sequences of remainder classes to the sequences of their representatives - just as conversely, by adding modules, one can turn a Cauchy network of integers into a sequence of remainder classes that make up the same pro-finite number.


  • The set of per-finite numbers is uncountable .
Commutative diagram for the ring of
perfinite numbers
  • The projective limit together with the homomorphisms
the canonical projections (of the projective Limes), has the following universal property :
For each group and homomorphisms for which applies to all , there is a clearly defined homomorphism so that applies.
Universal property of the embedding
  • The natural homomorphism has the following universal property :
For every homomorphism in a per- finite group there is a continuous homomorphism (with regard to the Krull topology) with
  • The isomorphism follows from the unambiguous prime factorization in
(with as the set of natural prime numbers ) from the direct product of the p -adic number rings to the projective limits
are. In the inverse function of isomorphism can be at any vector with components the archetype (unique) using the Chinese remainder theorem to determine the iterative in an extended process similar to that in the proof of the tightness in the article Limes (Category Theory) brought, applied is.
As in the projective Limes, addition and multiplication in the direct product take place component by component. This means that there are zero divisors in and can not have a quotient field .
For each prime number denote
the canonical projection (of the direct product). Applied to the injection
fulfills it The composition, on the other hand, corresponds to multiplication
  • A sequence of numbers converging in convergent also converges in every pro-finite subring and vice versa. The convergence for a single one is not enough, however. Example: the sequence in against converges, diverges both for prime numbers equal to and in Because is the order of the multiplicative group of the finite field, then for all and

The product topology on is the coarsest topology (the topology with the fewest open sets) with respect to which all projections are continuous.

This topology coincides with the Limes topology mentioned above and is called the Krull topology. Since the isomorphism that establishes isomorphism is simultaneously continuous in both directions under the topologies on both sides, it is also a homeomorphism .


The expansion of a perfinite number contains (like that of a real one ) an infinite number of symbols. The algorithms that process such symbol sequences can only process finite initial pieces. In the event of a termination, an indication of the magnitude of the error is desirable, similar to the p -adic numbers, for which the last digit thrown out is exact.

Representation as a direct product

The representation of a per-finite number as a direct product

is an infinite “vector” in two dimensions. In this representation, many algebraic number theoretic properties of are easily recognizable from the properties in the .

Representation as an infinite series

In the projective Limes , the partial order of the divisibility relation can be replaced by a linear order . To do this , put the “value” (the weight) in place and with the “base”. Then is

each element being an infinite family


of remainder classes. Each such representative can be calculated as a partial sum


write a series of “digits”     in place value notation with multiple bases .

The indexing is chosen in such a way that the digit represents a residual class - with an index higher by 1 - and the subsequent member represents a residual class of the "module" (at the point ).

In the induction hypothesis, let the digits of the representation be determined in such a way that

The requirement comes in the induction step

add the compatibility condition for all dividers

met with one of the canonical projections of the projective Limes. However, the already established congruences should be retained, i.e. H.

be valid. The extended Euclidean algorithm

supplies to the two modules and beside the greatest common divisor of two numbers with

Because similar to in

what put together

results. So can and

form so that with


as well as

applies as it should be. ■
The choice of shown leads to system A003418 of the least common multiple and to system A051451, while a choice with the
-fold module and any leads to the faculty-based system.

In every induction step, the algorithm combines two (simultaneous) congruences into a new one, which is equivalent to the two initial congruences, using the Chinese remainder theorem (with the aid of the extended Euclidean algorithm ). (In the case of non-prime modules, solvability is always guaranteed by the compatibility conditions of the projective system.) Regardless of the choice of the base system, the method throws out a sequence member of an infinite series per step.

Conversely, if digits   are chosen with   freely, then the infinite series formed with them and the given basic system represents a (unique) per- finite number .

Co-final episode

This series is only then a place value expansion for any arbitrary one if the given base system contains every prime number infinitely often, i.e. H. if the sequence of the modules is cofinal in and monotonic (increasing). This is the case with the system of faculties, the A003418 and A051451-based systems. The monotony avoids bases and is growing, since the interesting, open end of is with large numbers.


In the faculty-based number system (Engl. Factorial number system ) are as modules the faculties it and selected as bases. Lenstra gives for the sequence of symbols

–1 =… 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1
    = (… 1 0987654321) !

and denotes them with the subscript callsign. The number 1 is on the far left as in Lenstra Profinite Fibonacci numbers. P. 297 superscripted to express that it (possibly together with other superscript digits) up to and including the next normally written digit to the right of it belongs to a decimal number which makes up a single digit in the representation. The notation in the Horner scheme is:

    = (((((((((( 10 ) 10 + 9 ) 9 + 8 ) 8 + 7 ) 7 + 6 ) 6 + 5 ) 5 + 4 ) 4 + 3 ) 3 + 2 ) 2 + 1 ) 1
    = 11! - 1 = 39916799 ≡ -1 (mod 39916800 = 11!).

In this representation, per finite numbers, depending on their remainder mod 24 = 4 3 2, have the following developments in the first (rightmost) 3 digits:

xx (mod 24) 0 1 2 3 4th 5 6th 7th 8th 9 10 11 12 13 14th 15th 16 17th 18th ...
  ( z 3 z 2 z 1 ) !   000  001  010  011  020  021  100  101  110  111  120  121  200  201  210  211  220  221  300  ...

The choice of the faculties as modules in the faculty-based representation prefers the products of small prime factors , especially the prime factor 2.

A003418- or A051451-based

The following choice of bases and modules produces representations in which the natural numbers are preferred inversely proportional to their size.

Do this for each one first

( least common multiple ) the product of the maximum prime powers . Calculated in numbers, with

P : = ( P 1 , P 2 , P 3 , P 4 , P 5 , P 6 , P 7 , P 8 , P 9 , P 10 , ...)
= ( 1 , 1 · 2 = 2 , 2 3 = 6 , 6 2 = 12 , 12 5 = 60 , 60 1 = 60 , 60 7 = 420 , 420 2 = 840 , 840 3 = 2520 , 2520 · 1 = 2520 , ...)

the sequence A003418 in OEIS .

If one chooses for the representation as modules, then the corresponding bases are. Is not a prime power, then, however, is a prime power, for example then is a prime.

The example

-1 =… 10 1 0 3 2 2 1 7 6 1 0 5 4 2 1 3 2 2 1 1
    =… 1 0021604121,

in the Horner scheme

    = (((((((((( 10 ) 1 + 0 ) 3 + 2 ) 2 + 1 ) 7 + 6 ) 1 + 0 ) 5 + 4 ) 2 + 1 ) 3 + 2 ) 2 + 1 ) 1
    = P 12 - 1 = 27719 ≡ -1 (mod 27720 = P 12 ),

gives the representation of –1 (with only digits or with the digits in bold and the bases in normal print). The number 1 is on the far left as in Lenstra Profinite Fibonacci numbers. P. 297 in superscript to show that it belongs to the same position as the next normally written digit.

If one leaves out the bases = 1 together with the disappearing digits belonging to them, one has

to the modules
P 9 = 2520, P 8 = 840, P 7 = 420, P 5 = 60, P 4 = 12, P 3 = 6, P 2 = 2, P 1 = 1 
resp. to the bases
b 9 = 3,       b 8 = 2,       b 7 = 7,       b 5 = 5,       b 4 = 2,       b 3 = 3,       b 2 = 2        
the development
-1 = ... 1 6th  4th 
= ... 10 3 2 2 1 7 6 5 4 2 1 3 2 2 1 1
= ... 10 · P 9 +   2 · P 8 +   1 · P 7 +   6 · P 5 +   4 · P 4 +   1 · P 3 +   2 · P 2 +   1 
= ..., 27719, 2519, 839, 419, 59, 11, 5,
= ..., P 11 - 1, P 9 - 1, P 8 - 1, P 7 - 1, P 5 - 1, P 4 - 1, P 3 - 1, P 2 - 1 
≡ -1 (mod P n ) for all nN .

The modules P n of this representation make up the sequence A051451 in OEIS (with appropriately adapted indexing) .


Direct sum

The elements in the direct product , for which only a finite number of components differ from 0, are summarized in the direct sum

together. A profinite integer of this type can be used as - adic development of the form

with a base     and numbers   from   being written. It is said that the base is noted. The representation can be obtained from the representations with the Chinese remainder of the sentence .

The representation is clear and does not require a sign in front of the literal (the number constant) . For all bases is

All of these base representations are the same as in the ring

which is a subring of the direct sum.

From this representation it can be seen that (for one ) the base can be chosen without a square .

Prime powers

For every prime number and is

A family of residual classes from the projective Limes

fulfills the congruences for all


Congruences that

trivially imply. So it follows .

Conversely, if a family is from the projective Limes , then there are congruences for all

Fulfills. The families of residual classes

are a coarsening of the original families. And they meet the conditions . But since the sequence is co-final , they result in the same projective Limes. ■

The following consideration leads to the same result:
starting from the
-adic representation

with and you get directly to the subtotals


what about the -adic representation. This path can also be reversed - with the result:

10-adic numbers

The 10-adic numbers are an example of a -adic ring where the base is not a prime power. They are called the projective limes

and are a subring of the direct sum.


On the ring , yes, on the whole , an ultrametric can be defined, which turns into a metric space with the Krull topology.

A rational number can be written as with an integer and an assigned and prime. For each of them different from 0 there is a maximum exponent with this property. Analogously to , one function is completely defined as:
for   ,

The demands “non-negativity” and “positive definiteness” from the compilation of the amount function # amount function for bodies are easy to see. The "multiplicativity" cannot be fulfilled because it has zero divisors (see section #Zero divisors ). The " triangle inequality " results as follows: If the 2 numbers and different exponents and then the sum has the exponent But if they are the same, then with is so that the new exponent can by no means be smaller and the new amount cannot be larger. So it applies


Such a triangle inequality is called sharpened . The metric defined using this function

is thus an ultrametric . The topology it induces agrees with that defined by the filters .

10-adic to 2-adic and 5-adic

Is further     and respective representatives of the cosets then corresponds to the condition of the congruence

But it follows for

so that the same representatives make up both a pro-finite 2-adic number sequence and a per-finite 5-adic number sequence .

(2 × 5) -adic to 10-adic

Too freely chosen


is there a clearly defined with


Because the 2 simultaneous congruences


can be solved (unambiguously) for each with the Chinese remainder theorem because of the coprime nature of the modules .   is thereby   determined.

Zero divisor

Finite numbers (breaking number sequences) in the rings and are all in the ring of whole numbers. As is well known, the latter ring does not contain zero divisors , nor do the pro-finite rings and which have quotient fields, namely the 2-adic numbers or the 5-adic numbers

example 1

As explained in the section #Properties , the projection of a multiplication by are two different prime numbers, then is (component-wise multiplication in ). The product of two perfinite numbers can therefore be zero, even if both factors are different from zero.

The algorithm in the section Representation as Infinite Series yields for

to the status values 2520,  840,  420,  60,  12,  6,  2,  1
the A051451 development
1 2 = ... 1 · P 9 +   1 · P 8 +   0 · P 7 +   1 · P 5 +   3 · P 4 +   1 · P 3 +   1 · P 2 +   1
= ..., 3465, 945, 105, 105, 45, 9, 3, 1

The terms of the sequence in the last line are ≡1 (mod 2 n ) and divisible by (in the Limes always higher) powers of all other prime numbers.

The result is the A051451 development

1 5 = ... 8th · P 9 +   2 · P 8 +   0 · P 7 +   5 · P 5 +   3 · P 4 +   0 · P 3 +   0 · P 2 +   0
= ..., 22176, 2016, 336, 336, 36, 0, 0, 0

The terms of the sequence in the last line are ≡1 (mod 5 n ) and are divisible by increasingly higher powers of all other prime numbers.

The product of     the two sequences is divisible by increasing powers of 10 for increasing indices, i.e. H. it converges to 0 under the ultrametric .

Example 2

For be and . Because of

is divisor of . This means that the sequence converges in the ring of 10-adic numbers. Furthermore is . The same applies to .

The product is obviously divisible by arbitrarily high powers of 10, so that in

Incidentally, the two 10-adic numbers are idempotent because and have the consequence that and

Top rings

The ring of pro-finite rational numbers


includes , and is also


the ring of the finite Adele .

The product is the ring of the integer Adele.


  • Let be a prime number and the field with elements. Since every algebraic extension of cycle is of degree the Galois group to isomorphic is  , where the algebraic closure of means. The Frobenius automorphism corresponds
the producer of
  • the endomorphism ring of the module
  • In additive groups, per-finite multiples can be defined, in multiplicative per-finite exponents.

See also


Web links

References and comments

  1. In #Fried p. 14 called Prüfer group (German: Prüfergruppe ). ( See also Divisible Group )
  2. #Gille 3. The pro-finite completion of
  3. Proof in the article Limes (category theory)
  4. Nevertheless, there is no arrangement of which is compatible with the ring operations : The per-finite numbers cannot therefore be arranged. (This also applies to the p -adic numbers .)
  5. In the section pseudometrics # definition of a range by a uniform structure , starting from a uniform structure, here with the help of the countability of the fundamental system, a pseudometrics is constructed, which in turn induces. However, there is even a metric that induces the uniform structure :
    Be to it
    for   ,
    the “! value” of a . [ measures the proximity to zero (the degree of divisibility) of by divisors of the form ( pronounced: enn factorial ) - in analogy to the value in the rings that indicates the maximum exponent for divisibility by , or also to (see Lenstra Profinite number theory. p. 21 ) in the Archimedean systems.]
    Then applies to with
    with matching and from what The symmetrical case leads to Both cases result together
    The distance function thus formed
    fulfills the requirements for a metric and is an ultrametric :
    (1) Positive definiteness:   and  
    (2) symmetry:
    (3) Tightened triangle inequality:

    Like the uniform structure in the text, this metric is defined by the degree of divisibility, so that they match as uniform structures.

    NB: The episode is co-final in . And each monotonic cofinal sequence defines a metric with the same uniform structure.

  6. Because it is
  7. These zero nets are exactly the monotonic ones in cofinal nets, because
  8. #Brugger's Theorem 7.2.
  9. s. Article Limes (category theory)
  10. One implementation for this is the #Algorithm with the system A003418 of the least common multiple.
  11. a b The type of order occurring here is not but the type of order which is infinite in two dimensions (the sequence of prime numbers and the sequence of exponents)
    d. s. the vectors
      with   for almost everyone
    The order relation in goes component by component
    As a countable order type, it contains cofinal subsequences.
  12. Lenstra Prof Destinite number theory. P. 17
  13. provided that for every prime power there is a multiple among the place values,
  14. As usual with all place value notations , b -adic and p -adic, the small exponents are noted on the right side of the line. Most algorithms start there, especially addition and multiplication. The p -adic and the pro-finite numbers continue to the left towards the higher exponents, potentially to infinity.
  15. In contrast to the notations with the same base, the bases change from place to place, but depend on nothing but the number of the place. If they are also notated, they are as fixed as a scale division on a coordinate axis .
  16. This is in accordance with the convention for faculty-based number systems (also with Lenstra Profinite Groups Example 2.2).
  17. a b This sequence is strictly monotone cofinal in
  18. If the bases (or modules) are also noted, then the remainder classes to which the subtotals refer are also specified. This also applies to notations for which the bases are otherwise known or can be made accessible.
  19. Eric W. Weisstein "Smallest common multiple." From MathWorld - A Wolfram Web Resource
  20. This sequence is monotonically cofinal in
  21. Lenstra Profinite Groups Example 2.1
  22. The spelling is avoided in order not to evoke the association of a body .
  23. Fjelstad p. 11.
  24. It is true, however
  25. The place values ​​(or modules) are the weights by which the digits are to be multiplied, e.g. B. the number 3 with the accumulated weight 12 = 2 * 3 * 2 * 1.
  26. If one looks at this series as an integer sequence of numbers in the ring, then it is the same (if it converges against the one there) 1. It can also be understood as a sequence in , then it converges against (the one there) 0.
  27. Lenstra Prof Destinite number theory. P. 7
  28. Milne, Ch. I Example A. 5.
  29. Lenstra Profinite Fibonacci numbers. P. 299