# p-adic canonical form

Every non-zero $p$-adic number ($p$ is a positive rational prime number) can be uniquely written in canonical form^{}, formally as a Laurent series,

$$\xi ={a}_{-m}{p}^{-m}+{a}_{-m+1}{p}^{-m+1}+\mathrm{\cdots}+{a}_{0}+{a}_{1}p+{a}_{2}{p}^{2}+\mathrm{\cdots}$$ |

where $m\in \mathbb{N}$, $0\leqq {a}_{k}\leqq p-1$ for all $k$’s, and at least one of the integers ${a}_{k}$ is positive. In addition, we can write: $0=0+0p+0{p}^{2}+\mathrm{\cdots}$

The field ${\mathbb{Q}}_{p}$ of the $p$-adic numbers is the completion of the field $\mathbb{Q}$ with respect to its $p$-adic valuation^{} (http://planetmath.org/PAdicValuation); thus $\mathbb{Q}$ may be thought the subfield^{} (prime subfield^{}) of ${\mathbb{Q}}_{p}$. We can call the elements of ${\mathbb{Q}}_{p}\setminus \mathbb{Q}$ the proper $p$-adic numbers.

If, e.g., $p=2$, we have the 2-adic or, according to G. W. Leibniz, dyadic numbers, for which every ${a}_{k}$ is 0 or 1. In this case we can write the sum expression for $\xi $ in the reverse and use the ordinary positional (http://planetmath.org/Base3) (i.e., dyadic) figure system (http://planetmath.org/Base3). Then, for example, we have the rational numbers

$$-1=\mathrm{\dots}111111,$$ |

$$1=\mathrm{\dots}0001,$$ |

$$6.5=\mathrm{\dots}000110.1,$$ |

$$\frac{1}{5}=\mathrm{\dots}00110011001101.$$ |

(You may check the first by adding 1, and the last by multiplying by 5 = …000101.) All 2-adic rational numbers have periodic binary expansion (http://planetmath.org/DecimalExpansion). Similarly as the decimal (http://planetmath.org/DecimalExpansion) (according to Leibniz: decadic) expansions of irrational real numbers are aperiodic, the proper 2-adic numbers also have aperiodic binary expansion, for example the 2-adic fractional number

$$\alpha =\mathrm{\dots}1000010001001011.10111.$$ |

The 2-adic fractional numbers have some bits “1” after the dyadic point “.” (in continental Europe: comma “$,$”), the 2-adic integers have none. The 2-adic integers form a subring of the 2-adic field ${\mathbb{Q}}_{2}$ such that ${\mathbb{Q}}_{2}$ is the quotient field of this ring.

Every such 2-adic integer $\epsilon $ whose last bit is “1”, as $-3/7=\mathrm{\dots}11011011011$, is a unit of this ring, because the division $1:\epsilon $ clearly gives as quotient a integer (by the way, the divisions of the binary expansions in practice go from right to left and are very comfortable!).

Those integers ending in a “0” are non-units of the ring, and they apparently form the only maximal ideal^{} in the ring (which thus is a local ring^{}). This is a principal ideal^{} $\U0001d52d$, the generator of which may be taken $\mathrm{\dots}00010=10$ (i.e., two). Indeed, two is the only prime number of the ring, but it has infinitely many associates^{}, a kind of copies, namely all expansions of the form $\mathrm{\dots}10=\epsilon \cdot 10$. The only non-trivial ideals in the ring of 2-adic integers are $\U0001d52d,{\U0001d52d}^{2},{\U0001d52d}^{3},\mathrm{\dots}$ They have only 0 as common element.

All 2-adic non-zero integers are of the form $\epsilon \cdot {2}^{n}$ where $n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots}$. The values $n=-1,-2,-3,\mathrm{\dots}$ here would give non-integral, i.e. fractional 2-adic numbers.

If in the binary of an arbitrary 2-adic number, the last non-zero digit “1” corresponds to the power ${2}^{n}$, then the 2-adic valuation of the 2-adic number $\xi $ is given by

$${|\xi |}_{2}={2}^{-n}.$$ |

Title | p-adic canonical form |

Canonical name | PadicCanonicalForm |

Date of creation | 2013-03-22 14:13:37 |

Last modified on | 2013-03-22 14:13:37 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 66 |

Author | pahio (2872) |

Entry type | Example |

Classification | msc 12F99 |

Related topic | IntegralElement |

Related topic | UltrametricTriangleInequality |

Related topic | NonIsomorphicCompletionsOfMathbbQ |

Related topic | IdealsOfADiscreteValuationRingArePowersOfItsMaximalIdeal |

Defines | proper p-adic number |

Defines | dyadic number |

Defines | dyadic point |

Defines | 2-adic fractional number |

Defines | 2-adic integer |

Defines | 2-adic valuation |