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:  $0=0+0p+0p^2+\mathrm{\cdots }$

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 $𝔭$, 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  $𝔭,{𝔭}^2,{𝔭}^3,\mathrm{\dots }$  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