# 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 11 subcategories, out of 11 total.

### O

### P

### R

### T

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

The following 66 pages are in this category, out of 66 total.

### C

### E

- Equivalence Class in P-adic Integers Contains Unique Coherent Sequence
- Equivalence Class in P-adic Integers Contains Unique P-adic Expansion
- Equivalence Class in P-adic Numbers Contains Unique P-adic Expansion
- Equivalence of Definitions of Convergent P-adic Sequence
- Equivalence of Definitions of P-adic Norms

### 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 P-adic Integers/Unit Ball
- 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

### L

### P

- P-adic Expansion Converges to P-adic Number
- P-adic Expansion is a Cauchy Sequence in P-adic Norm
- P-adic Expansion is a Cauchy Sequence in P-adic Norm/Converges to P-adic Number
- P-adic Expansion is a Cauchy Sequence in P-adic Norm/Represents a P-adic Number
- 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 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 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 satisfies Non-Archimedean Norm Axioms
- P-adic Norms are Not Equivalent
- P-adic Number is Limit of Unique P-adic Expansion
- 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 Numbers form Non-Archimedean Valued Field
- 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