# p-adic Numbers

The set **Q**_{p} of p-adic numbers is the completion of the space of all rationals **Q** when the latter is endowed with a special metric |.|_{p}. Not only **Q**_{p} is different from **R** (giving the first example of the dependency of the completion on the underlying metric), but **Q**_{p}, being defined for every prime p, supplies a sequence of distinct completions of the same set of the rationals.

First, for every prime p, we define norm |.|_{p}, on the set **Q** of rational numbers. Every nonzero rational number

q = ±p^{n}p_{1}^{n1}p_{2}^{n2} ·...· p_{m}^{nm} = p^{n} (u / v),

where p and p_{k} are distinct primes and the exponents n and n_{k} may be positive or negative, depending on the prevalence of the power of the corresponsing prime in the numerator or denominator of q; u and v are integers prime to p.

### Definition

|q|_{p} = p^{-n},

where n is the exponent of p in the above factorization. Separately we also define _{p} = 0.

|1001/1000 - 1|_{2} = |1/1000|_{2} = |2^{-3}5^{-3}|_{2} = 8.

Which shows that 1001/1000 and 1 are not at all close in **Q**_{2}. In **Q**_{5}, the situation is even worse: _{5} = 125._{p} ≤ 1.

|.|_{p} has three important properties that qualify it as a *norm*:

1. | |u|_{p} = 0 only if u = 0. | |

2. | |uv|_{p} = |u|_{p}|v|_{p}, for any rational u and v | |

3. | |u + v|_{p} ≤ |u|_{p} + |v|_{p}, for any rational u and v |

In fact the triangle inequality in 3. can be strengthened

3'. | |u + v|_{p} ≤ max{|u|_{p}, |v|_{p}}, for any rational u and v. |

An even stronger result holds:

3''. | For |u|_{p} ≠ |v|_{p}, |u + v|_{p} = max{|u|_{p}, |v|_{p}}, for any rational u and v. |

To prove the latter, assume |u|_{p} < |v|_{p}. Then, by 3', _{p} ≤ |v|_{p}._{p} = |u + v - u|_{p} ≤ max{|u + v|_{p}, |u|_{p}} = |u + v|_{p},

Norm |.|_{p} is called the *p-adic norm* (or *p-adic absolute value*). It was introduced by Kurt Hensel, a student of Leopold Kronecker, in 1902.

The availability of the triangle inequality leads to a definition of a metric _{p}(u, v) = |u - v|_{p}**Q**_{p} of **Q**. The members of **Q**_{p} are known as *p-adic* numbers. p-adic numbers u that satisfy _{p} ≤ 1**Z**_{p} (not to be confused with the field of residues modulo p which uses the same notation.) The set of integers **Z** is dense in **Z**_{p}, meaning that every p-adic integer is a limit (in the p-norm of an integer sequence.) Any (reduced) fraction whose denominator contains no factors of p belongs to **Z**_{p}. p is the only prime p-adic integer.

It is easy to give an example of an integer sequence which is Cauchy in the sense of the p-adic norm. The motivation comes from ^{n}|_{p} = p^{-n}.

x_{0} | = 1, | |

x_{1} | = 1 + p, | |

x_{2} | = 1 + p + p², | |

x_{3} | = 1 + p + p² + p³, | |

... | ||

_{n} | = 1 + p + p² + ... + p^{n}, | |

... |

For the sequence {x_{k}}, assuming n > m,

|x_{n} - x_{m}|_{p} | = |p^{m+1} + ... + p^{n}|_{p} | |

= p^{-(m+1)} |

which tends to 0 as m (and hence n) grows. What p-adic number does {x_{k}} converge to? This is a particular case of what is known as a p-adic expansion.

### References

- E. B. Berger,
*Exploring the Number Jungle*, AMS, 2000 - J. R. Goldman,
*The Queen of Mathematics*, A K Peters, 1998 - F. Q. Gouvêa,
__Local and Global in Number Theory__, in*The Princeton Companion to Mathematics*T. Gowers (ed.), Princeton University Press, 2008

- p-adic Numbers
- p-adic Expansions
- The Local-Global Principle
- Triangulating Squares
- Square Root of 2 is Irrational

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Copyright © 1996-2018 Alexander BogomolnyStrange as it may appear, we'll write

q | = 1 + p + ... + p^{m} + ... |

As we just said, q is well defined as a limit of the sequence {x_{k}} and this is how the identity should be understood. All arithmetic operations are permitted to carry over the limits so that

q | = 1 + p + ... + p^{m} + ... | |

pq | = p + p² + ... + p^{m+1} + ... | |

= 1 + p + ... + p^{m} + ... = q |

So that pq + 1 = q, implying q = 1 / (1 - p). Taking p = 2 we obtain:

-1 | = 1 + 2 + 4 + ... |

One certainly needs time and effort to get used to such weirdness.

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