# Pro-finite number

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 ${\ displaystyle {\ widehat {\ mathbb {Z}}}}$ ${\ displaystyle \ mathbb {Z}}$

${\ displaystyle \ iota \ colon \ quad {\ begin {array} {lll} \ mathbb {Z} & \ hookrightarrow & {\ widehat {\ mathbb {Z}}} \\ r & \ mapsto & (r + 1 \ mathbb {Z}, r + 2 \ mathbb {Z}, r + 3 \ mathbb {Z}, \ dots) \\\ end {array}}}$

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${\ displaystyle r: = 1 \ in \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} / i \ mathbb {Z}}$${\ displaystyle 1 + i \ mathbb {Z}}$${\ displaystyle (1 + 1 \ mathbb {Z}, 1 + 2 \ mathbb {Z}, 1 + 3 \ mathbb {Z}, \ dots)}$${\ displaystyle {\ widehat {\ mathbb {Z}}}.}$

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. ${\ displaystyle \ mathbb {Z}}$${\ displaystyle {\ widehat {\ mathbb {Z}}}.}$${\ displaystyle {\ widehat {\ mathbb {Z}}}}$

The Galois group of the algebraic closure of a finite field over this field is isomorphic to${\ displaystyle {\ widehat {\ mathbb {Z}}}.}$

## definition

The perfinite completion of the group of integers is ${\ displaystyle \ mathbb {Z}}$

${\ displaystyle {\ widehat {\ mathbb {Z}}}: = \ varprojlim _ {i \ in \ mathbb {N}} \ mathbb {Z} / i \ mathbb {Z}}$           (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") ${\ displaystyle {\ widehat {\ mathbb {Z}}}}$ ${\ displaystyle (\ mathbb {N}, \ mid),}$ ${\ displaystyle k \! \ mid \! j}$ ${\ displaystyle \ mathbb {Z} / j \ mathbb {Z}.}$${\ displaystyle j}$${\ displaystyle k \! \ mid \! j}$

${\ displaystyle f_ {jk} \ colon \ quad {\ begin {array} {lll} \ mathbb {Z} / j \ mathbb {Z} & \ to & \ mathbb {Z} / k \ mathbb {Z} \\ r + j \ mathbb {Z} & \ mapsto & r + k \ mathbb {Z}, \ end {array}}}$

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 ${\ displaystyle j \ mathbb {Z} \ subset k \ mathbb {Z}}$ ${\ displaystyle \ mathbb {Z} / j \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} / k \ mathbb {Z}}$${\ displaystyle k \! \ mid \! j \! \ mid \! i}$

${\ displaystyle f_ {ik} (r + i \ mathbb {Z}) = r + k \ mathbb {Z} = f_ {jk} (r + j \ mathbb {Z}) = f_ {jk} (f_ {ij } (r + i \ mathbb {Z}))}$ so ${\ displaystyle f_ {ik} = f_ {jk} \ circ f_ {ij},}$

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: ${\ displaystyle \ textstyle \ left (r_ {i} + i \ mathbb {Z} \ right) _ {i \ in \ mathbb {N}} \ in \ prod _ {i \ in \ mathbb {N}} \ mathbb {Z} / i \ mathbb {Z}}$${\ displaystyle j, k}$${\ displaystyle k \! \ mid \! j}$

${\ displaystyle f_ {jk} (r_ {j} + j \ mathbb {Z}) = r_ {k} + k \ mathbb {Z},}$

what by the congruences

 ${\ displaystyle r_ {j}}$ ${\ displaystyle \ equiv}$ ${\ displaystyle r_ {k} \, ({\ text {mod}} k)}$ ${\ displaystyle ({\ text {V}})}$

is fulfilled. Written in a formula it results:

${\ displaystyle {\ begin {array} {rrlll} {\ widehat {\ mathbb {Z}}} &: = && \ displaystyle \ varprojlim _ {i \ in \ mathbb {N}} \ mathbb {Z} / i \ mathbb {Z} \\ & = & \ displaystyle {\ Big \ {} \ left (r_ {i} + i \ mathbb {Z} \ right) _ {i \ in \ mathbb {N}} \ in & \ displaystyle \ prod _ {i \ in \ mathbb {N}} \ mathbb {Z} / i \ mathbb {Z} \; \; {\ Big |} \; \; \ forall j, k \ in \ mathbb {N} \,: \, k \! \ mid \! j \ Rightarrow r_ {j} \ equiv r_ {k} \, ({\ text {mod}} k) {\ Big \}} & \ quad ({\ text {pL}}) \\\ end {array}}}$

A family of elements that meet the compatibility requirements, which is part of the projective Limes, is sometimes also referred to as "fiber". ${\ displaystyle ({\ text {V}})}$

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. ${\ displaystyle {\ widehat {\ mathbb {Z}}}}$${\ displaystyle {\ widehat {\ mathbb {Z}}}}$

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 . ${\ displaystyle {\ widehat {\ mathbb {Z}}}}$${\ displaystyle \ mathbb {Z} / i \ mathbb {Z}}$${\ displaystyle {\ widehat {\ mathbb {Z}}}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ Pi _ {i \ in \ mathbb {N}} (\ mathbb {Z} / i \ mathbb {Z}),}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle {\ widehat {\ mathbb {Z}}}}$

## Alternative construction

The ring of integers can one also in "classic" style uniform structure completed be. Be for this${\ displaystyle \ mathbb {Z}}$ ${\ displaystyle N \ in \ mathbb {N}}$

${\ displaystyle U_ {N}: = \ {(x, y) \ mid xy \ in N \ mathbb {Z} \} = \ {(x, y) \ mid x \ equiv y \, ({\ text { mod}} N) \}}$

a neighborhood (of order ). The set is a (countable) fundamental system and the associated filter ${\ displaystyle N}$${\ displaystyle F: = \ {U_ {N} \ mid N \ in \ mathbb {N} \}}$

${\ displaystyle \ Phi: = \ {V \ mid \ exists N \ in \ mathbb {N}: U_ {N} \ subset V \}}$

a uniform structure for . The demands on are easily verified: ${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ Phi}$

(1) Each neighborhood and each contains the diagonal${\ displaystyle U_ {N}}$${\ displaystyle V \ in \ Phi}$${\ displaystyle \ {(x, x) \ mid x \ in \ mathbb {Z} \}.}$
(2) Is and , then is${\ displaystyle V \ in \ Phi}$${\ displaystyle V \ subset W}$${\ displaystyle W \ in \ Phi.}$
(3) Is , then is also${\ displaystyle V, W \ in \ Phi}$${\ displaystyle V \ cap W \ in \ Phi.}$
(4) For each there is a with .${\ displaystyle V \ in \ Phi}$${\ displaystyle W \ in \ Phi}$${\ displaystyle W ^ {2} \ subset V}$
(5) Is , then is also${\ displaystyle V \ in \ Phi}$${\ displaystyle V ^ {- 1} \ in \ Phi.}$

The Amount of Cauchy nets in is ${\ displaystyle {\ mathcal {C}}}$${\ displaystyle (\ mathbb {Z}, \ Phi)}$

${\ displaystyle {\ begin {array} {rrlll} {\ mathcal {C}} &: = & {\ Big \ {} \ left (r_ {j} \ right) _ {j \ in \ mathbb {N}} \ in \ prod _ {j \ in \ mathbb {N}} \ mathbb {Z} \; \; {\ Big |} \; \; \ forall N \ in \ mathbb {N} \; \ exists n_ {N } \ in \ mathbb {N} \ ,: n_ {N} \ mid j, k \ Rightarrow (r_ {j}, r_ {k}) \ in U_ {N} {\ Big \}} & \ quad \; ({\ text {CN}}), \\\ end {array}}}$

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 ). ${\ displaystyle \ mathbb {Z}}$ ${\ displaystyle {\ mathcal {C}} / {\ mathcal {N}}}$ ${\ displaystyle {\ mathcal {N}}}$${\ displaystyle 0}$

${\ displaystyle {\ mathcal {C}} / {\ mathcal {N}}}$ turns out to be isomorphic to ${\ displaystyle {\ widehat {\ mathbb {Z}}}.}$

 proof Be a family of compatible residual classes, so ${\ displaystyle \ textstyle \ left (r_ {i} + i \ mathbb {Z} \ right) _ {i \ in \ mathbb {N}} \ in \ prod _ {i \ in \ mathbb {N}} \ mathbb {Z} / i \ mathbb {Z}}$ ${\ displaystyle \ forall j, k: k \! \ mid \! j \ implies r_ {j} \ equiv r_ {k} \, ({\ text {mod}} k)}$, and be , then is for everyone with${\ displaystyle N \ in \ mathbb {N}}$${\ displaystyle j, k \ in \ mathbb {N}}$${\ displaystyle N \! \ mid \! j, k}$ ${\ displaystyle r_ {j} \ equiv r_ {N} \ equiv r_ {k} \, ({\ text {mod}} N)}$, the consequence of the representatives is a Cauchy network. ${\ displaystyle \ textstyle \ left (r_ {i} \ right) _ {i \ in \ mathbb {N}}}$ 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 ${\ displaystyle \ textstyle \ left (s _ {\ nu} \ right) _ {\ nu \ in \ mathbb {N}}}$${\ displaystyle k \ in \ mathbb {N}}$${\ displaystyle \ nu _ {k} \ in \ mathbb {N}}$${\ displaystyle \ nu, \ mu \ in \ mathbb {N}}$${\ displaystyle \ nu _ {k} \! \ mid \! \ nu, \ mu}$ ${\ displaystyle s _ {\ mu} \ equiv s _ {\ nu} \, ({\ text {mod}} k)}$ . If you take it now , then it is ${\ displaystyle \ nu: = \ nu _ {k}}$ ${\ displaystyle s _ {\ mu} \ equiv s _ {\ nu _ {k}} \, ({\ text {mod}} k)}$ 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 ${\ displaystyle \ mu \ in \ mathbb {N}}$${\ displaystyle \ nu _ {k} \! \ mid \! \ mu}$${\ displaystyle \ textstyle \ left (r_ {k} \ right) _ {k \ in \ mathbb {N}}: = \ left (s _ {\ nu _ {k}} \ right) _ {k \ in \ mathbb {N}}}$${\ displaystyle \ in {\ mathcal {C}} / {\ mathcal {N}}}$${\ displaystyle k \ in \ mathbb {N}}$${\ displaystyle j \ in \ mathbb {N}}$${\ displaystyle k \! \ mid \! j}$${\ displaystyle \ nu _ {k} \! \ mid \! \ nu _ {j}}$${\ displaystyle s _ {\ nu _ {j}} \ equiv s _ {\ nu _ {k}} \, ({\ text {mod}} k)}$ ${\ displaystyle r_ {j} \ equiv r_ {k} \, ({\ text {mod}} k)}$ . The sequence of residual classes thus fulfills the compatibility requirements and is a family ${\ displaystyle \ textstyle \ left (r_ {k} + k \ mathbb {Z} \ right) _ {k \ in \ mathbb {N}}}$${\ displaystyle ({\ text {V}})}$ ${\ displaystyle \ in \ varprojlim _ {k \ in \ mathbb {N}} \ mathbb {Z} / k \ mathbb {Z} = {\ widehat {\ mathbb {Z}}}}$ .
Result
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.${\ displaystyle r_ {j} + j \ mathbb {Z}}$ ${\ displaystyle r_ {j}}$

## properties

• The set of per-finite numbers is uncountable .${\ displaystyle {\ widehat {\ mathbb {Z}}}}$
• ${\ displaystyle {\ widehat {\ mathbb {Z}}}}$is a per-finite group .
Commutative diagram for the ring of
perfinite numbers
• The projective limit together with the homomorphisms${\ displaystyle {\ widehat {\ mathbb {Z}}}}$
${\ displaystyle \ tau _ {i} \ colon \ quad {\ begin {array} {lll} {\ widehat {\ mathbb {Z}}} & \ to & \ mathbb {Z} / i \ mathbb {Z} \ \\ left (x_ {n} \ right) _ {n \ in \ mathbb {N}} & \ mapsto & x_ {i}, \ end {array}}}$
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.${\ displaystyle T}$${\ displaystyle t_ {i} \ colon T \ to \ mathbb {Z} / i \ mathbb {Z},}$${\ displaystyle t_ {j} = f_ {ij} \ circ t_ {i}}$${\ displaystyle j \! \ mid \! i}$${\ displaystyle t \ colon T \ to {\ widehat {\ mathbb {Z}}},}$${\ displaystyle t_ {i} = \ tau _ {i} \ circ t}$
Universal property of the embedding
isomorphism${\ displaystyle \ iota}$
• The natural homomorphism has the following universal property :${\ displaystyle \ iota \ colon \ mathbb {Z} \ to {\ widehat {\ mathbb {Z}}}}$
For every homomorphism in a per- finite group there is a continuous homomorphism (with regard to the Krull topology) with${\ displaystyle f \ colon \ mathbb {Z} \ to G}$${\ displaystyle G}$${\ displaystyle {\ hat {f}} \ colon {\ widehat {\ mathbb {Z}}} \ to G}$${\ displaystyle f = {\ hat {f}} \ circ \ iota.}$
• The isomorphism follows from the unambiguous prime factorization in${\ displaystyle \ mathbb {Z}}$
${\ displaystyle {\ widehat {\ mathbb {Z}}} \ cong \ prod _ {p \ in \ mathbb {P}} \ mathbb {Z} _ {p}}$
(with as the set of natural prime numbers ) from the direct product of the p -adic number rings to the projective limits ${\ displaystyle \ mathbb {P}}$${\ displaystyle {\ widehat {\ mathbb {Z}}}}$ ${\ displaystyle \ mathbb {Z} _ {p},}$
${\ displaystyle \ mathbb {Z} _ {p} = \ varprojlim _ {n \ in \ mathbb {N}} \ mathbb {Z} / p ^ {n} \ mathbb {Z}}$
are. In the inverse function of isomorphism can be at any vector with components the archetype (unique) using the Chinese remainder theorem to determine the iterative in an extended process similar to that in the proof of the tightness in the article Limes (Category Theory) brought, applied is.${\ displaystyle \ left (x_ {p} \ right) _ {p \ in \ mathbb {P}}}$${\ displaystyle x_ {p} \ in \ mathbb {Z} _ {p}}$${\ displaystyle x \ in {\ widehat {\ mathbb {Z}}}}$
As in the projective Limes, addition and multiplication in the direct product take place component by component. This means that there are zero divisors in and can not have a quotient field .${\ displaystyle {\ widehat {\ mathbb {Z}}}}$${\ displaystyle {\ widehat {\ mathbb {Z}}}}$
For each prime number denote ${\ displaystyle p \ in \ mathbb {P}}$
${\ displaystyle {\ begin {array} {llll} \ pi _ {p} \ colon & {\ widehat {\ mathbb {Z}}} & \ to & \ mathbb {Z} _ {p} \\\ end { array}}}$
the canonical projection (of the direct product). Applied to the injection
 ${\ displaystyle \ iota _ {p} \ colon}$ ${\ displaystyle \ mathbb {Z} _ {p}}$ ${\ displaystyle \ to}$ ${\ displaystyle {\ widehat {\ mathbb {Z}}} \; = \; \ textstyle \ Pi _ {p \ in \ mathbb {P}} \ mathbb {Z} _ {p}}$ ${\ displaystyle z}$ ${\ displaystyle \ mapsto}$ ${\ displaystyle (0, \ ldots, 0, z}$ ${\ displaystyle, 0, \ ldots)}$ ${\ displaystyle \ uparrow}$ component ${\ displaystyle \ mathbb {Z} _ {p}}$
fulfills it The composition, on the other hand, corresponds to multiplication${\ displaystyle \ pi _ {p} \ circ \ iota _ {p} = \ operatorname {id} \ mid _ {\ mathbb {Z} _ {p}}.}$ ${\ displaystyle \ iota _ {p} \ circ \ pi _ {p}}$
 ${\ displaystyle \ cdot 1_ {p} \ colon}$ ${\ displaystyle {\ widehat {\ mathbb {Z}}} = \ mathbb {Z} _ {2} \ times \ mathbb {Z} _ {3} \ times \ ldots}$ ${\ displaystyle \ to}$ ${\ displaystyle {\ widehat {\ mathbb {Z}}} \; = \; \ textstyle \ Pi _ {p \ in \ mathbb {P}} \ mathbb {Z} _ {p}}$ ${\ displaystyle x = (x_ {2}, x_ {3}, x_ {5}, \ ldots, x_ {p}, \ ldots)}$ ${\ displaystyle \ mapsto}$ ${\ displaystyle (0, \ ldots, 0, x_ {p}}$ ${\ displaystyle, 0, \ ldots)}$ ${\ displaystyle = x \ cdot 1_ {p}}$ With ${\ displaystyle 1_ {p}}$ ${\ displaystyle: =}$ ${\ displaystyle (0, \ ldots, 0, \; 1}$ ${\ displaystyle, 0, \ ldots)}$ ${\ displaystyle \ in {\ widehat {\ mathbb {Z}}}}$ ${\ displaystyle \ uparrow}$ component ${\ displaystyle \ mathbb {Z} _ {p}}$
• 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${\ displaystyle {\ widehat {\ mathbb {Z}}}}$${\ displaystyle \ mathbb {Z} _ {p}}$${\ displaystyle p \ in \ mathbb {P}}$${\ displaystyle \ left (2 ^ {n} \ right) _ {n \ in \ mathbb {N}},}$${\ displaystyle \ mathbb {Z} _ {2}}$${\ displaystyle 0}$${\ displaystyle \ mathbb {Z} _ {p}}$${\ displaystyle p}$${\ displaystyle 2}$${\ displaystyle {\ widehat {\ mathbb {Z}}}.}$${\ displaystyle k: = \ operatorname {ord} _ {p} (2)}$${\ displaystyle 2 = 1 + 1}$ ${\ displaystyle (\ mathbb {Z} / p \ mathbb {Z}) ^ {\ times}}$${\ displaystyle n \ in \ mathbb {Z}: \; 2 ^ {kn} \ equiv 1 \, ({\ text {mod}} p)}$${\ displaystyle 2 ^ {kn + 1} \ equiv 2 \ not \ equiv 1 \, ({\ text {mod}} p).}$
topology

The product topology on is the coarsest topology (the topology with the fewest open sets) with respect to which all projections are continuous. ${\ displaystyle {\ widehat {\ mathbb {Z}}}}$${\ displaystyle \ pi _ {p}}$

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 .

## presentation

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 ${\ displaystyle x \ in {\ widehat {\ mathbb {Z}}}}$

${\ displaystyle x = \ left (x_ {p} \ right) _ {p \ in \ mathbb {P}} \ qquad \ in \ textstyle \ Pi _ {p \ in \ mathbb {P}} \ mathbb {Z} _ {p}}$

is an infinite “vector” in two dimensions. In this representation, many algebraic number theoretic properties of are easily recognizable from the properties in the . ${\ displaystyle x}$${\ displaystyle \ mathbb {Z} _ {p}}$

### 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${\ displaystyle ({\ text {pL}})}$${\ displaystyle (\ mathbb {N}, \ mid)}$${\ displaystyle (\ mathbb {N}, <)}$${\ displaystyle m_ {n} \ in \ mathbb {Z}}$${\ displaystyle m_ {1} = 1}$${\ displaystyle n}$${\ displaystyle b_ {n} \ in \ mathbb {N}}$${\ displaystyle m_ {n-1} b_ {n} = m_ {n}}$

${\ displaystyle {\ widehat {\ mathbb {Z}}} = {\ Big \ {} \ left (x_ {n} \ right) _ {n \ in \ mathbb {N}} \ in \ prod _ {n \ in \ mathbb {N}} \ mathbb {Z} / m_ {n} \ mathbb {Z} \; \; {\ Big |} \; \; \ forall n \ in \ mathbb {N}: x_ {n + 1} \ equiv x_ {n} \, ({\ text {mod}} m_ {n}) {\ Big \}},}$

each element being an infinite family ${\ displaystyle x = \ left (x_ {n} \ right) _ {n \ in \ mathbb {N}}}$

 ${\ displaystyle x}$ ${\ displaystyle =:}$ ${\ displaystyle \ left (\ mathbb {Z} \, m_ {n + 1} \; + \ right.}$ ${\ displaystyle \ left.r_ {n} \ right) _ {n \ in \ mathbb {N}}}$

of remainder classes. Each such representative can be calculated as a partial sum ${\ displaystyle r_ {n} \ in \ mathbb {Z}}$

 ${\ displaystyle r_ {n}}$ ${\ displaystyle =}$ ${\ displaystyle \ textstyle \ sum _ {i = 1} ^ {n}}$ ${\ displaystyle z_ {i}}$ ${\ displaystyle \ cdot m_ {i}}$ ${\ displaystyle =}$ ${\ displaystyle z_ {n}}$ ${\ displaystyle \ cdot m_ {n}}$ ${\ displaystyle + \ dots + z_ {3}}$ ${\ displaystyle \ cdot m_ {3}}$ ${\ displaystyle + z_ {2}}$ ${\ displaystyle \ cdot m_ {2}}$ ${\ displaystyle + z_ {1} \; \ cdot \! 1}$ ${\ displaystyle =}$ ${\ displaystyle ((}$ ${\ displaystyle z_ {n}}$ ${\ displaystyle \ cdot b_ {n}}$ ${\ displaystyle + \ dots + z_ {3})}$ ${\ displaystyle \ cdot b_ {3}}$ ${\ displaystyle + z_ {2})}$ ${\ displaystyle \ cdot b_ {2}}$ ${\ displaystyle + z_ {1}}$ ${\ displaystyle ({\ text {R}} _ {n})}$

write a series of “digits”     in place value notation with multiple bases .${\ displaystyle ({\ text {R}})}$${\ displaystyle z_ {n} \ in \ mathbb {Z} \; \; \ wedge \; \; 0 \ leq z_ {n}

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 ).${\ displaystyle z_ {n}}$${\ displaystyle {\ text {mod}} b_ {n + 1}}$${\ displaystyle r_ {n}}$${\ displaystyle {\ text {mod}} m_ {n + 1},}$${\ displaystyle n}$

algorithm
In the induction hypothesis, let the digits of the representation be determined in such a way that ${\ displaystyle j \ in \ {1,2, \ dots, n-1 \}}$${\ displaystyle z_ {j} \ in \ mathbb {Z}}$
 ${\ displaystyle r_ {j}}$ ${\ displaystyle \ equiv \ textstyle \ sum _ {i = 1} ^ {j} z_ {i} \ cdot m_ {i}}$ ${\ displaystyle ({\ text {mod}} m_ {j + 1})}$ ${\ displaystyle ({\ text {R}} _ {j})}$

The requirement comes in the induction step

 ${\ displaystyle r_ {n}}$ ${\ displaystyle \ equiv q_ {n}}$ ${\ displaystyle ({\ text {mod}} n)}$

add the compatibility condition for all dividers${\ displaystyle g \ mid n}$

 ${\ displaystyle q_ {n}}$ ${\ displaystyle \ equiv \ tau _ {g} (x)}$ ${\ displaystyle ({\ text {mod}} g)}$ ${\ displaystyle ({\ text {V}} _ {g})}$

met with one of the canonical projections of the projective Limes. However, the already established congruences should be retained, i.e. H. ${\ displaystyle \ tau}$

 ${\ displaystyle r_ {n}}$ ${\ displaystyle \ equiv r_ {n-1}}$ ${\ displaystyle ({\ text {mod}} m_ {n})}$

be valid. The extended Euclidean algorithm

 ${\ displaystyle (g, u, v)}$ ${\ displaystyle: = \ operatorname {extended \ _euclid} (m_ {n}, n)}$

supplies to the two modules and beside the greatest common divisor of two numbers with ${\ displaystyle m_ {n}}$${\ displaystyle n}$ ${\ displaystyle g}$${\ displaystyle u, v \ in \ mathbb {Z}}$

 ${\ displaystyle g}$ ${\ displaystyle = u \ cdot m_ {n} + v \ cdot n.}$

Because similar to in${\ displaystyle g \ mid m_ {n}}$${\ displaystyle ({\ text {V}} _ {g})}$

 ${\ displaystyle r_ {n-1}}$ ${\ displaystyle \ equiv \ tau _ {g} (x)}$ ${\ displaystyle ({\ text {mod}} g),}$

what put together

 ${\ displaystyle q_ {n}}$ ${\ displaystyle \ equiv r_ {n-1}}$ ${\ displaystyle ({\ text {mod}} g)}$

results. So can and ${\ displaystyle (q_ {n} -r_ {n-1}) / g}$

 ${\ displaystyle z_ {n}}$ ${\ displaystyle: = (q_ {n} -r_ {n-1}) / g \ cdot u}$

form so that with

 ${\ displaystyle r_ {n}}$ ${\ displaystyle: = z_ {n} \ cdot m_ {n} + r_ {n-1}}$ ${\ displaystyle ({\ text {R}} _ {n})}$

either

 ${\ displaystyle r_ {n}}$ ${\ displaystyle \ equiv r_ {n-1}}$ ${\ displaystyle ({\ text {mod}} m_ {n})}$

as well as

 ${\ displaystyle r_ {n}}$ ${\ displaystyle \ equiv (q_ {n} -r_ {n-1}) / g \ cdot (u \ cdot m_ {n}) + r_ {n-1}}$ ${\ displaystyle \ equiv (q_ {n} -r_ {n-1}) / g \ cdot (gv \ cdot n) + r_ {n-1}}$ ${\ displaystyle \ equiv q_ {n}}$ ${\ displaystyle ({\ text {mod}} n)}$

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. ${\ displaystyle z_ {n}}$${\ displaystyle z_ {n}: = u \ cdot (q_ {n} -r_ {n-1}) / g + k \ cdot n / g \, ({\ text {mod}} n)}$${\ displaystyle g}$${\ displaystyle n}$${\ displaystyle k \ in \ mathbb {Z}}$

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 . ${\ displaystyle z_ {n} \ in \ mathbb {Z} \; \; \ wedge \; \; 0 \ leq z_ {n} ${\ displaystyle \ left (b_ {n} \ right) _ {n \ in \ mathbb {N}}}$${\ displaystyle ({\ text {R}})}$

#### 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. ${\ displaystyle x \ in {\ widehat {\ mathbb {Z}}}}$${\ displaystyle \ left (b_ {n} \ right) _ {n \ in \ mathbb {N}}}$${\ displaystyle \ left (m_ {n} \ right) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle (\ mathbb {N}, |)}$${\ displaystyle b_ {n} <1}$${\ displaystyle (\ mathbb {N}, |)}$

#### Faculty-based

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 ${\ displaystyle m_ {n}: = n!}$${\ displaystyle b_ {n}: = n}$${\ displaystyle -1 \ in {\ widehat {\ mathbb {Z}}}}$

–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 ${\ displaystyle n \ in \ mathbb {N}}$

${\ displaystyle P_ {n}: = \ operatorname {kgV} (1,2, \ dots, n)}$

( least common multiple ) the product of the maximum prime powers . Calculated in numbers, with ${\ displaystyle \ leq n}$

 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. ${\ displaystyle m_ {n}: = P_ {n}}$${\ displaystyle b_ {n} = m_ {n} / m_ {n-1}.}$${\ displaystyle n}$${\ displaystyle b_ {n} = 1.}$${\ displaystyle n> 1}$${\ displaystyle p ^ {k},}$${\ displaystyle b_ {n} = p}$

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 0 2 1 6th 4th 1 2 1 = ... 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, 1 = ..., 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 n ∈ N .

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

## Sub-rings

### 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${\ displaystyle \ Pi _ {p \ in \ mathbb {P}} \ mathbb {Z} _ {p}}$

${\ displaystyle \ bigoplus _ {p \ in \ mathbb {P}} \ mathbb {Z} _ {p}: = {\ Big \ {} \ left (x_ {p} \ right) _ {p \ in \ mathbb {P}} \ in \ prod _ {p \ in \ mathbb {P}} \ mathbb {Z} _ {p} \; {\ Big |} \; \ exists n \ in \ mathbb {N} \, \ forall p \ in \ mathbb {P} \,: \, p> n \ Rightarrow x_ {p} = 0 {\ Big \}}}$

together. A profinite integer of this type can be used as - adic development of the form ${\ displaystyle \ textstyle x \ in \ bigoplus _ {p \ in \ mathbb {P}} \ mathbb {Z} _ {p}}$${\ displaystyle b}$

${\ displaystyle x = \ sum _ {i = 0} ^ {\ infty} r_ {i} \ cdot b ^ {i}}$

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 . ${\ displaystyle \ textstyle b: = \ prod _ {p \ in \ mathbb {P}} ^ {x_ {p} \ neq 0} p}$${\ displaystyle r_ {i}}$${\ displaystyle \ {0,1, \ ldots, b-1 \}}$${\ displaystyle x}$${\ displaystyle b}$${\ displaystyle b}$${\ displaystyle p}$

The representation is clear and does not require a sign in front of the literal (the number constant) . For all bases is ${\ displaystyle b \ in \ mathbb {Z} \ setminus \ {- 1,0,1 \}}$

${\ displaystyle -1 = \ sum _ {i = 0} ^ {\ infty} (b-1) \ cdot b ^ {i}.}$

All of these base representations are the same as in the ring ${\ displaystyle b}$

${\ displaystyle \ mathbb {Z} _ {b} = \ varprojlim _ {n \ in \ mathbb {N}} \ mathbb {Z} / b ^ {n} \ mathbb {Z} \ cong \ prod _ {p \ in \ mathbb {P}} ^ {p \ mid b} \ mathbb {Z} _ {p},}$

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 . ${\ displaystyle \ textstyle x \ in \ bigoplus _ {p \ in \ mathbb {P}} \ mathbb {Z} _ {p}}$${\ displaystyle b}$

### Prime powers

For every prime number and is ${\ displaystyle p}$${\ displaystyle m \ in \ mathbb {N}}$

${\ displaystyle \ mathbb {Z} _ {p ^ {m}} = \ varprojlim _ {i \ in \ mathbb {N}} \ mathbb {Z} / p ^ {mi} \ mathbb {Z} = \ varprojlim _ {i \ in \ mathbb {N}} \ mathbb {Z} / p ^ {i} \ mathbb {Z} = \ mathbb {Z} _ {p}}$ .
proof
A family of residual classes from the projective Limes${\ displaystyle \ textstyle \ left (r_ {i} + p ^ {mi} \ mathbb {Z} \ right) _ {i \ in \ mathbb {N}} \ in \ prod _ {i \ in \ mathbb {N }} \ mathbb {Z} / p ^ {mi} \ mathbb {Z}}$${\ displaystyle \ varprojlim _ {i \ in \ mathbb {N}} \ mathbb {Z} / p ^ {mi} \ mathbb {Z} = \ mathbb {Z} _ {p ^ {m}}}$

fulfills the congruences for all${\ displaystyle j \ geq k}$

 ${\ displaystyle r_ {j}}$ ${\ displaystyle \ equiv}$ ${\ displaystyle r_ {k} \, ({\ text {mod}} p ^ {mk})}$ ${\ displaystyle ({\ text {V}} _ {p ^ {m}})}$ ,

Congruences that

 ${\ displaystyle r_ {j}}$ ${\ displaystyle \ equiv}$ ${\ displaystyle r_ {k} \, ({\ text {mod}} p ^ {k})}$ ${\ displaystyle ({\ text {V}} _ {p})}$

trivially imply. So it follows . ${\ displaystyle \ textstyle \ left (r_ {i} + p ^ {mi} \ mathbb {Z} \ right) _ {i \ in \ mathbb {N}} \ in \ varprojlim _ {i \ in \ mathbb {N }} \ mathbb {Z} / p ^ {i} \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} _ {p ^ {m}} \ subset \ varprojlim _ {i \ in \ mathbb {N}} \ mathbb {Z} / p ^ {i} \ mathbb {Z} = \ mathbb {Z} _ {p}}$

Conversely, if a family is from the projective Limes , then there are congruences for all${\ displaystyle \ textstyle \ left (r_ {i} + p ^ {i} \ mathbb {Z} \ right) _ {i \ in \ mathbb {N}} \ in \ prod _ {i \ in \ mathbb {N }} \ mathbb {Z} / p ^ {i} \ mathbb {Z} = \ mathbb {Z} _ {p}}$${\ displaystyle \ varprojlim _ {i \ in \ mathbb {N}} \ mathbb {Z} / p ^ {i} \ mathbb {Z}}$${\ displaystyle j \ geq k}$

 ${\ displaystyle r_ {j}}$ ${\ displaystyle \ equiv}$ ${\ displaystyle r_ {k} \, ({\ text {mod}} p ^ {k})}$ ${\ displaystyle ({\ text {V}} _ {p})}$

Fulfills. The families of residual classes

 ${\ displaystyle \ textstyle \ left (r_ {i} + p ^ {m \ left \ lfloor {\ frac {i} {m}} \ right \ rfloor} \ mathbb {Z} \ right) _ {i \ in \ mathbb {N}}}$

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. ■ ${\ displaystyle ({\ text {V}} _ {p ^ {m}})}$${\ displaystyle \ textstyle \ left (p ^ {m \ left \ lfloor {\ frac {i} {m}} \ right \ rfloor} \ right) _ {i \ in \ mathbb {N}}}$${\ displaystyle \ textstyle \ left (p ^ {i} \ right) _ {i \ in \ mathbb {N}}}$

The following consideration leads to the same result:
starting from the
-adic representation ${\ displaystyle p}$

${\ displaystyle x = \ sum _ {i = k} ^ {\ infty} r_ {i} p ^ {i}}$

with and you get directly to the subtotals${\ displaystyle k \ in \ mathbb {Z}}$${\ displaystyle r_ {i} \ in \ {0, \ dots, p-1 \}}$${\ displaystyle \ textstyle s_ {j}: = \ sum _ {i = 0} ^ {m-1} r_ {mj + i} \, p ^ {i}}$

${\ displaystyle x = \ sum _ {j = \ left \ lfloor {\ frac {k} {m}} \ right \ rfloor} ^ {\ infty} s_ {j} p ^ {mj}}$ ,

what about the -adic representation. This path can also be reversed - with the result: ${\ displaystyle s_ {j} \ in \ {0, \ dots, p ^ {m} -1 \}}$${\ displaystyle p ^ {m}}$

${\ displaystyle \ mathbb {Z} _ {p ^ {m}} = \ mathbb {Z} _ {p}}$

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 ${\ displaystyle b}$${\ displaystyle b = 10 = 2 \ cdot 5}$

${\ displaystyle \ mathbb {Z} _ {10} = \ varprojlim _ {i \ in \ mathbb {N}} \ mathbb {Z} / 10 ^ {i} \ mathbb {Z}}$

and are a subring of the direct sum.

#### Ultrametric

On the ring , yes, on the whole , an ultrametric can be defined, which turns into a metric space with the Krull topology. ${\ displaystyle \ mathbb {Z} _ {10}}$${\ displaystyle \ mathbb {Q} _ {2} \ times \ mathbb {Q} _ {5}}$ ${\ displaystyle d_ {10}}$${\ displaystyle \ mathbb {Q} _ {2} \ times \ mathbb {Q} _ {5}}$

proof
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:${\ displaystyle r \ in \ mathbb {Q} ^ {\ times}}$${\ displaystyle r = \ pm {\ tfrac {p} {q}} 10 ^ {s}}$${\ displaystyle p, q, s}$${\ displaystyle 10}$${\ displaystyle p}$${\ displaystyle q.}$${\ displaystyle r}$${\ displaystyle s}$${\ displaystyle \ mathbb {Q} _ {p}}$${\ displaystyle \ mathbb {Q} _ {2} \ times \ mathbb {Q} _ {5}}$${\ displaystyle \ psi}$
 ${\ displaystyle \ psi (r): = \; {\ begin {cases} \\\\\ end {cases}}}$ ${\ displaystyle 0}$ for   , ${\ displaystyle r = 0}$ ${\ displaystyle {\ text {e}} ^ {- s}}$ otherwise.

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 ${\ displaystyle \ mathbb {Z} _ {10}}$${\ displaystyle r}$${\ displaystyle r '}$${\ displaystyle s}$${\ displaystyle s',}$${\ displaystyle \ min (s, s').}$${\ displaystyle r + r '= \ pm {\ tfrac {pq' + p'q} {qq '}} 10 ^ {s}}$${\ displaystyle \ operatorname {ggT} (qq ', 10) = 1,}$

${\ displaystyle \ psi (r + r ') \ leq \ max (\ psi (r), \ psi (r')).}$  ■

Such a triangle inequality is called sharpened . The metric defined using this function${\ displaystyle \ psi}$

${\ displaystyle d_ {10} (x, y): = \ psi (xy)}$

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

Is further     and respective representatives of the cosets then corresponds to the condition of the congruence${\ displaystyle \ left (x_ {n} \ right) _ {n \ in \ mathbb {N}} \ in \ mathbb {Z} _ {10},}$${\ displaystyle j \ leq i,}$${\ displaystyle r_ {i}, r_ {j} \ in \ mathbb {Z}}$${\ displaystyle x_ {i} = r_ {i} + 10 ^ {i} \ mathbb {Z}, x_ {j} = r_ {j} + 10 ^ {j} \ mathbb {Z},}$${\ displaystyle r_ {j} + 10 ^ {j} \ mathbb {Z} = f_ {ij} (r_ {i} + 10 ^ {i} \ mathbb {Z})}$

${\ displaystyle r_ {j} \ equiv r_ {i} \, ({\ text {mod}} 10 ^ {j}).}$

But it follows for ${\ displaystyle p \ in \ {2.5 \}}$

${\ displaystyle r_ {j} \ equiv r_ {i} \, ({\ text {mod}} p ^ {j}),}$

so that the same representatives make up both a pro-finite 2-adic number sequence and a per-finite 5-adic number sequence . ${\ displaystyle r_ {n} \ in \ mathbb {Z}}$${\ displaystyle \ left (r_ {n} \ right) _ {n \ in \ mathbb {N}} \ in \ mathbb {Z} _ {2}}$${\ displaystyle \ left (r_ {n} \ right) _ {n \ in \ mathbb {N}} \ in \ mathbb {Z} _ {5}}$

Too freely chosen

 ${\ displaystyle \ textstyle y =: \ sum _ {j = 0} ^ {\ infty} y_ {j} 2 ^ {j} \ in \ mathbb {Z} _ {2}}$ and ${\ displaystyle \ textstyle z =: \ sum _ {j = 0} ^ {\ infty} z_ {j} 5 ^ {j} \ in \ mathbb {Z} _ {5}}$

is there a clearly defined with ${\ displaystyle x = \ left (x_ {i} \ right) _ {i \ in \ mathbb {N}} \ in \ mathbb {Z} _ {10},}$

 ${\ displaystyle \ pi _ {2} (x) = y}$ and ${\ displaystyle \ pi _ {5} (x) = z}$ ${\ displaystyle ({\ text {sK}} _ {\ infty}).}$

Because the 2 simultaneous congruences

 ${\ displaystyle \ textstyle x_ {i} \ equiv \ sum _ {j = 0} ^ {i-1} y_ {j} 2 ^ {j} \, ({\ text {mod}} 2 ^ {i}) }$ and ${\ displaystyle \ textstyle x_ {i} \ equiv \ sum _ {j = 0} ^ {i-1} z_ {j} 5 ^ {j} \, ({\ text {mod}} 5 ^ {i}) }$ ${\ displaystyle ({\ text {sK}} _ {i})}$

can be solved (unambiguously) for each with the Chinese remainder theorem because of the coprime nature of the modules .   is thereby   determined. ${\ displaystyle i}$${\ displaystyle \ textstyle x_ {i}}$${\ displaystyle \ equiv \, ({\ text {mod}} \ operatorname {kgV} (2 ^ {i}, 5 ^ {i}) = 10 ^ {i})}$

#### 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${\ displaystyle \ mathbb {Z} _ {2}, \ mathbb {Z} _ {5}}$${\ displaystyle \ mathbb {Z} _ {10}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} _ {2}}$${\ displaystyle \ mathbb {Z} _ {5},}$ ${\ displaystyle \ mathbb {Q} _ {2}}$${\ displaystyle \ mathbb {Q} _ {5}.}$

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. ${\ displaystyle p \ in \ mathbb {P}}$${\ displaystyle \ pi _ {p}}$${\ displaystyle 1_ {p}.}$${\ displaystyle p, q}$${\ displaystyle 1_ {p} \ cdot 1_ {q} = 0}$${\ displaystyle \ textstyle \ Pi _ {p \ in \ mathbb {P}} \ mathbb {Z} _ {p}}$

The algorithm in the section Representation as Infinite Series yields for${\ displaystyle 1_ {2}}$

 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 ${\ displaystyle 1_ {5}}$

 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 . ${\ displaystyle 1_ {2} \ cdot 1_ {5}}$${\ displaystyle d_ {10}}$

Example 2

For be and . Because of ${\ displaystyle n \ in \ mathbb {N} _ {0}}$${\ displaystyle x_ {n}: = 6 ^ {5 ^ {n}}}$${\ displaystyle y_ {n}: = 5 ^ {2 ^ {n}}}$

${\ displaystyle {\ begin {array} {rlll} (x_ {n + 2} -x_ {n + 1}) & / \; (x_ {n + 1} -x_ {n}) \\ = ({x_ {n}} ^ {5 \ cdot 5} - \; \; {x_ {n}} ^ {5}) & / \; ({x_ {n}} ^ {5} -x_ {n}) & = {x_ {n}} ^ {4 \ cdot 5} & + {x_ {n}} ^ {4 \ cdot 4} & + {x_ {n}} ^ {4 \ cdot 3} & + {x_ {n} } ^ {4 \ cdot 2} & + {x_ {n}} ^ {4 \ cdot 1} \\ = (6 ^ {5 ^ {n + 2}} - 6 ^ {5 ^ {n + 1}} ) & / \; (6 ^ {5 ^ {n + 1}} - 6 ^ {5 ^ {n}}) & = (6 ^ {5 ^ {n}}) ^ {4 \ cdot 5} & + (6 ^ {5 ^ {n}}) ^ {4 \ cdot 4} & + (6 ^ {5 ^ {n}}) ^ {4 \ cdot 3} & + (6 ^ {5 ^ {n}} ) ^ {4 \ cdot 2} & + (6 ^ {5 ^ {n}}) ^ {4 \ cdot 1} \\ && \ equiv \; \; 6 & + \; \; 6 & + \; \; 6 & + \; \; 6 & + \; \; 6 \; \; = 5 \ cdot 6 \\ && \ equiv 0 &&& ({\ text {mod}} 10) \ end {array}}}$

is divisor of . This means that the sequence converges in the ring of 10-adic numbers. Furthermore is . The same applies to . ${\ displaystyle 10 ^ {n}}$${\ displaystyle x_ {n} -x_ {n-1}}$${\ displaystyle x: = \ lim _ {n \ to \ infty} x_ {n}}$${\ displaystyle x \ equiv 6 \ not \ equiv 0 \; ({\ text {mod}} 10)}$${\ displaystyle y: = \ lim _ {n \ to \ infty} y_ {n} \ equiv 5 \ not \ equiv 0 \; ({\ text {mod}} 10)}$

The product is obviously divisible by arbitrarily high powers of 10, so that in${\ displaystyle x \ cdot y = \ lim _ {n \ to \ infty} x_ {n} \ cdot y_ {n}}$${\ displaystyle x \ cdot y = 0}$${\ displaystyle \ mathbb {Z} _ {10}.}$

Incidentally, the two 10-adic numbers are idempotent because and have the consequence that and${\ displaystyle x_ {n + 1} = x_ {n} ^ {\, 5}}$${\ displaystyle y_ {n + 1} = y_ {n} ^ {\, 2}}$${\ displaystyle x ^ {5} = x}$${\ displaystyle y ^ {2} = y.}$

## Top rings

The ring of pro-finite rational numbers

 ${\ displaystyle {\ widehat {\ mathbb {Q}}}}$ ${\ displaystyle: =}$ ${\ displaystyle \ textstyle {\ Big \ {} \ left (x_ {i} \ right) _ {i \ in \ mathbb {N}} \ in \ prod _ {i \ in \ mathbb {N}} \ mathbb { Q} / i \ mathbb {Z} \; \; {\ Big |} \; \; \ forall j, k \ in \ mathbb {N} \,: \, k \! \ Mid \! J \ Rightarrow x_ {j} \ equiv x_ {k} \, ({\ text {mod}} k) {\ Big \}}}$

includes , and is also ${\ displaystyle {\ widehat {\ mathbb {Z}}}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Q}.}$

 ${\ displaystyle {\ widehat {\ mathbb {Q}}}}$ ${\ displaystyle =}$ ${\ displaystyle \ mathbb {Q} + {\ widehat {\ mathbb {Z}}} \; = \; \ mathbb {Q} \ cdot {\ widehat {\ mathbb {Z}}} \; \ cong \; \ mathbb {Q} \ otimes _ {\ mathbb {Z}} {\ widehat {\ mathbb {Z}}}}$ ${\ displaystyle \ cong}$ ${\ displaystyle \ textstyle {\ Big \ {} \ left (x_ {p} \ right) _ {p \ in \ mathbb {P}} \ in \ prod _ {p \ in \ mathbb {P}} \ mathbb { Q} _ {p} \; \; {\ Big |} \; \; \ exists n \ in \ mathbb {N} \, \ forall p \ in \ mathbb {P} \,: \, p> n \ Rightarrow x_ {p} \ in \ mathbb {Z} _ {p} {\ Big \}}}$

The product is the ring of the integer Adele. ${\ displaystyle {\ widehat {\ mathbb {Z}}} \ times \ mathbb {R}}$

## Applications

• 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${\ displaystyle p}$${\ displaystyle \ mathbb {F} _ {p ^ {n}}}$${\ displaystyle {p ^ {n}}}$ ${\ displaystyle \ mathbb {F} _ {p ^ {n}}}$${\ displaystyle \ mathbb {F} _ {p}}$ ${\ displaystyle n,}$${\ displaystyle \ mathbb {Z} / n \ mathbb {Z},}$${\ displaystyle \ operatorname {Gal} ({\ overline {\ mathbb {F}}} _ {p} / \ mathbb {F} _ {p}) = {\ widehat {\ mathbb {Z}}}}$${\ displaystyle {\ overline {\ mathbb {F}}} _ {p}}$${\ displaystyle \ mathbb {F} _ {p}}$
${\ displaystyle {\ mathcal {F}} _ {p} \ colon {\ begin {array} {lll} {\ overline {\ mathbb {F}}} _ {p} & \ to & {\ overline {\ mathbb {F}}} _ {p} \\ x & \ mapsto & x ^ {p} \ end {array}}}$
the producer of${\ displaystyle (1,1, \ dots)}$${\ displaystyle {\ widehat {\ mathbb {Z}}}.}$
• ${\ displaystyle {\ widehat {\ mathbb {Z}}} = \ operatorname {End} (\ mathbb {Q} / \ mathbb {Z}),}$the endomorphism ring of the module${\ displaystyle \ mathbb {Q} / \ mathbb {Z}.}$
• In additive groups, per-finite multiples can be defined, in multiplicative per-finite exponents.

## literature

1. In #Fried p. 14 called Prüfer group (German: Prüfergruppe ). ( See also Divisible Group )
2. #Gille 3. The pro-finite completion of${\ displaystyle \ mathbb {Z}}$
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 .)${\ displaystyle {\ widehat {\ mathbb {Z}}}}$
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 : ${\ displaystyle \ Phi,}$${\ displaystyle \ Phi}$
${\ displaystyle \ Phi}$
Be to it
 ${\ displaystyle v (x): = \; {\ begin {cases} \\\\\ end {cases}}}$ ${\ displaystyle + \ infty}$ for   , ${\ displaystyle x = 0}$ ${\ displaystyle \ max \ {n \ in \ mathbb {N} \, \ mid \, n! | x \}}$ otherwise.
the “! value” of a . [ measures the proximity to zero (the degree of divisibility) of by divisors of the form ( pronounced: enn factorial ) - in analogy to the value in the rings that indicates the maximum exponent for divisibility by , or also to (see Lenstra Profinite number theory. p. 21 ) in the Archimedean systems.]${\ displaystyle x \ in \ mathbb {Z}}$${\ displaystyle v}$${\ displaystyle x}$${\ displaystyle n!}$${\ displaystyle p}$${\ displaystyle \ mathbb {Z} _ {p},}$${\ displaystyle n}$${\ displaystyle p ^ {n}}$${\ displaystyle \ varepsilon \ to 0}$
Then applies to with${\ displaystyle x, y \ in \ mathbb {Z}}$${\ displaystyle v (x) \ leq v (y)}$
${\ displaystyle x = \ xi v (x)!, \ quad y = \ eta v (y)!}$
with matching and from what The symmetrical case leads to Both cases result together ${\ displaystyle \ xi, \ eta \ in \ mathbb {Z}}$${\ displaystyle x + y = v (x)! (\ xi + \ eta v (y)! / v (x)!),}$${\ displaystyle v (x + y) \ geq v (x).}$${\ displaystyle v (x) \ geq v (y)}$${\ displaystyle v (x + y) \ geq v (y).}$
${\ displaystyle v (x + y) \ geq \ min \ {v (x), v (y) \}.}$
The distance function thus formed
${\ displaystyle {\ text {d}} \ left (x, y \ right): = {\ text {e}} ^ {- v (xy)}}$
fulfills the requirements for a metric and is an ultrametric :
 (1) Positive definiteness: ${\ displaystyle {\ text {d}} \ left (x, y \ right) \ geq 0}$   and   ${\ displaystyle {\ text {d}} \ left (x, y \ right) = 0 \ Leftrightarrow x = y}$ (2) symmetry: ${\ displaystyle {\ text {d}} \ left (x, y \ right) = {\ text {d}} (y, x)}$ (3) Tightened triangle inequality: ${\ displaystyle {\ text {d}} (x, y) \ leq \ max \ {{\ text {d}} (x, z), {\ text {d}} (z, y) \}}$

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. ${\ displaystyle \ left (n! \ right) _ {n \ in \ mathbb {N}}}$${\ displaystyle (\ mathbb {N}, |)}$

6. Because it is${\ displaystyle U_ {N} ^ {2} = U_ {N}.}$
7. These zero nets are exactly the monotonic ones in cofinal nets, because ${\ displaystyle \ left (r_ {n} \ right) _ {n \ in \ mathbb {N}}}$${\ displaystyle (\ mathbb {N}, |)}$
${\ displaystyle {\ begin {array} {ll} & \ forall N \ in \ mathbb {N} \; \ exists n \ in \ mathbb {N} \ ,: N \ mid r_ {n} \ quad \ wedge \ quad \ forall n \ in \ mathbb {N}: r_ {n} \ mid r_ {n + 1} \\\ Rightarrow & \ displaystyle \ lim _ {\ infty \ leftarrow n} r_ {n} = 0 \\\ Rightarrow & \ forall N \ in \ mathbb {N} \; \ exists n_ {N} \ in \ mathbb {N} \ ,: n_ {N} \ mid n \ Rightarrow N \ mid r_ {n} \\\ Rightarrow & \ forall N \ in \ mathbb {N} \; \ exists n \ in \ mathbb {N} \ ,: N \ mid r_ {n} \\\ end {array}}}$
8. #Brugger's Theorem 7.2.
9. s. Article Limes (category theory)
10. One implementation for this is the #Algorithm with the system A003418 of the least common multiple.
11. a b The type of order occurring here is not but the type of order which is infinite in two dimensions (the sequence of prime numbers and the sequence of exponents) ${\ displaystyle \ omega = (\ mathbb {N}, <),}$
${\ displaystyle (\ mathbb {N}, \ mid) = (\ mathbb {N} _ {0}, \ leq) ^ {(\ mathbb {P})} = \ bigoplus _ {p \ in \ mathbb {P }} p ^ {\ mathbb {N} _ {0}},}$
d. s. the vectors
${\ displaystyle n = \ Pi p ^ {n_ {p}} \ colon \ mathbb {P} \ to \ mathbb {N} _ {0}}$   with   for almost everyone${\ displaystyle n_ {p} = 0}$${\ displaystyle p.}$
The order relation in goes component by component ${\ displaystyle (\ mathbb {N}, \ mid)}$
${\ displaystyle n \ mid m \ quad \ Longleftrightarrow \ quad \ forall p \ in \ mathbb {P}: n_ {p} \ leq m_ {p}.}$
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. ${\ displaystyle \ forall N \ in \ mathbb {N} \; \ exists n \ in \ mathbb {N} \ ,: N \ mid m_ {n}}$
18. ${\ displaystyle \ forall n \ in \ mathbb {N}: m_ {n} \ mid m_ {n + 1}}$
19. a b This sequence is strictly monotone cofinal in${\ displaystyle (\ mathbb {N}, |).}$
20. 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.
21. mathworld.wolfram.com Eric W. Weisstein "Smallest common multiple." From MathWorld - A Wolfram Web Resource
22. This sequence is monotonically cofinal in${\ displaystyle (\ mathbb {N}, |).}$
23. Lenstra Profinite Groups Example 2.1
24. The spelling is avoided in order not to evoke the association of a body .${\ displaystyle \ mathbb {Q} _ {10}}$
26. It is true, however${\ displaystyle \ psi (r \ cdot r ') \ leq \ psi (r) \ cdot \ psi (r').}$
28. 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.${\ displaystyle \ mathbb {Z} _ {2},}$${\ displaystyle \ mathbb {Z} _ {3}, \ mathbb {Z} _ {5}, \ ldots}$