p-adic numbers
E483405
The p-adic numbers are a system of number fields that extend the rational numbers by measuring distance with respect to divisibility by a fixed prime p, playing a central role in modern number theory and arithmetic geometry.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| p-adic Numbers, p-adic Analysis, and Zeta-Functions | 1 |
| p-adic numbers Q_p | 1 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
complete valued field
ⓘ
field ⓘ local field ⓘ non-Archimedean field ⓘ number system ⓘ topological field ⓘ |
| completionOf | rational numbers with respect to the p-adic norm ⓘ |
| constructedFrom |
p-adic absolute value
ⓘ
p-adic metric ⓘ |
| contrastsWith | real numbers ⓘ |
| definedOver | rational numbers ⓘ |
| denotedBy | Q_p NERFINISHED ⓘ |
| distanceMeasures | divisibility by p rather than size ⓘ |
| extends | rational numbers ⓘ |
| formsPartOf | product decomposition of adeles ⓘ |
| generalizes | Hensel’s lemma applications ⓘ |
| hasAbsoluteValue | p-adic absolute value ⓘ |
| hasBasisOfNeighborhoodsOfZero | powers of p ⓘ |
| hasCharacteristic | 0 ⓘ |
| hasElementRepresentation | infinite series in powers of p with coefficients 0,…,p-1 ⓘ |
| hasPrime | p ⓘ |
| hasSubring |
Z_p
ⓘ
p-adic integers ⓘ |
| hasTopologyInducedBy | p-adic metric ⓘ |
| hasValuation | p-adic valuation ⓘ |
| introducedBy | Kurt Hensel NERFINISHED ⓘ |
| introducedIn | 1897 ⓘ |
| isCompleteWithRespectTo | p-adic metric ⓘ |
| isLocallyCompact | true ⓘ |
| isNonArchimedean | true ⓘ |
| isTotallyDisconnected | true ⓘ |
| maximalIdealGeneratedBy | p ⓘ |
| parameterizedBy | prime number p ⓘ |
| playsCentralRoleIn | local analysis of arithmetic problems ⓘ |
| relatedTo |
Hasse principle
NERFINISHED
ⓘ
adeles ⓘ ideles ⓘ |
| residueField | finite field F_p ⓘ |
| satisfies | ultrametric inequality ⓘ |
| usedIn |
Diophantine equations
NERFINISHED
ⓘ
Galois representations NERFINISHED ⓘ Iwasawa theory NERFINISHED ⓘ algebraic number theory ⓘ arithmetic geometry ⓘ local class field theory ⓘ local-global principles ⓘ number theory ⓘ p-adic Hodge theory NERFINISHED ⓘ |
| valuationRing | p-adic integers ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
p-adic Numbers, p-adic Analysis, and Zeta-Functions
this entity surface form:
p-adic numbers Q_p