# Category:P-adic Number Theory

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This category contains results about **P-adic Number Theory**.

Definitions specific to this category can be found in Definitions/P-adic Number Theory.

**P-adic number theory** is the branch of mathematics which studies the properties of the $p$-adic numbers.

## Subcategories

This category has the following 20 subcategories, out of 20 total.

## Pages in category "P-adic Number Theory"

The following 71 pages are in this category, out of 71 total.

### C

### E

### I

- Image of P-adic Norm
- Integer Arbitrarily Close to Rational in Valuation Ring of P-adic Norm
- Integers are Arbitrarily Close to P-adic Integers
- Integers are Dense in P-adic Integers
- Integers are Dense in Unit Ball of P-adic Numbers
- Integers form Subring of P-adic Integers
- Integers form Subring of Valuation Ring of P-adic Norm on Rationals

### P

- P-adic Closed Ball is Instance of Closed Ball of a Norm
- P-adic Expansion Converges to P-adic Number
- P-adic Expansion Converges to P-adic Number iff P-adic Expansion Represents P-adic Number
- P-adic Expansion is a Cauchy Sequence in P-adic Norm
- P-adic Expansion Less Intial Zero Terms Represents Same P-adic Number
- P-adic Expansion Representative of P-adic Number is Unique
- P-adic Expansion Represents a P-adic Number
- P-adic Integer has Unique Coherent Sequence Representative
- P-adic Integer has Unique P-adic Expansion Representative
- P-adic Integer is Limit of Unique Coherent Sequence of Integers
- P-adic Integer is Limit of Unique P-adic Expansion
- P-adic Integers is Metric Completion of Integers
- P-adic Norm and Absolute Value are Not Equivalent
- P-adic Norm forms Non-Archimedean Valued Field
- P-adic Norm is Well Defined
- P-adic Norm not Complete on Rational Numbers
- P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient
- P-adic Norm of p-adic Number is Power of p
- P-adic Norm on Rational Numbers Forms Non-Archimedean Valued Field
- P-adic Norm on Rational Numbers is Non-Archimedean Norm
- P-adic Norm satisfies Non-Archimedean Norm Axioms
- P-adic Norms are Not Equivalent
- P-adic Number has Unique P-adic Expansion Representative
- P-adic Number is Limit of Unique P-adic Expansion
- P-adic Number is Power of p Times P-adic Unit
- P-adic Number times Integer Power of p is P-adic Integer
- P-adic Number times P-adic Norm is P-adic Unit
- P-adic Numbers are Generated Ring Extension of P-adic Integers
- P-adic Numbers are Uncountable
- P-adic Numbers form Completion of Rational Numbers with P-adic Norm
- P-adic Open Ball is Instance of Open Ball of a Norm
- P-adic Sphere is Instance of Sphere of a Norm
- P-adic Valuation Extends to P-adic Numbers
- Partial Sums of P-adic Expansion forms Coherent Sequence
- Product Formula for Norms on Non-zero Rationals
- Product Formula for Norms on Non-zero Rationals/Lemma

### R

- Rational Numbers are Dense Subfield of P-adic Numbers
- Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit
- Representative of P-adic Number is Representative of Equivalence Class
- Representatives of same P-adic Number iff Difference is Null Sequence
- Residue Field of P-adic Norm on Rationals

### S

- Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers
- Sequence of Consecutive Integers Modulo Power of p is Cauchy in P-adic Norm
- Set of P-adic Integers is Clopen in P-adic Numbers
- Sphere is Set Difference of Closed and Open Ball in P-adic Numbers
- Sphere is Set Difference of Closed Ball with Open Ball/P-adic Numbers
- Summary of Topology on P-adic Numbers