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# 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,

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

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: $0=0+0p+0p^{2}+\cdots$

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; 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 order and use the ordinary positional binary (i.e., dyadic) figure system. Then, for example, we have the rational numbers

$-1=...111111,$ |

$1=...0001,$ |

$6.5=...000110.1,$ |

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

(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. Similarly as the decimal (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

$\alpha=...1000010001001011.10111.$ |

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 $\varepsilon$ whose last bit is “1”, as $-3/7=...11011011011$, is a unit of this ring, because the division $1\colon\!\varepsilon$ clearly gives as quotient a similar 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 $\mathfrak{p}$, the generator of which may be taken $...00010=10$ (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 $...10=\varepsilon\cdot 10$. The only non-trivial ideals in the ring of 2-adic integers are $\mathfrak{p},\,\mathfrak{p}^{2},\,\mathfrak{p}^{3},\,\ldots$ They have only 0 as common element.

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

If in the binary representation 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

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

## Mathematics Subject Classification

12F99*no label found*

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