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The p-adic integers are the compatible residue-class tuples in the inverse limit of Z mod p^n
Definition
Fix a prime . The -adic integers are
where the transition maps are reduction modulo . Concretely, is the set of tuples with and
so its elements are exactly the compatible tuples of The inverse limit is the set of compatible tuples in the Cartesian product.
With the inverse-limit topology, is a pro- group in the sense of A pro-p group is a profinite group that is an inverse limit of finite p-groups.
Depends on
Used by
- The canonical map from Z to Zp sends an integer to its coherent residue classes modulo pⁿ Definition
- The p-adic metric on Zp is determined by the first coordinate at which two compatible residue systems differ Definition
- A p-adic integer is encoded by a compatible sequence of residue digits Example
- Coordinatewise addition and negation make Zp a topological abelian group Lemma
- The additive group of Zp is torsion-free Proposition
- The canonical map from the integers into Zp is injective and has dense image Theorem
- The inverse-limit topology on Zp agrees with the p-adic metric topology Theorem
Dependency tree · two levels
5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Jordan Bell, The profinite completion of the integers, the p-adic integers, and Prufer p-groups (standard reference, not scraped)
- Gareth Wilkes, Profinite Groups and Group Cohomology lecture notes (standard reference, not scraped)