p-adic canonical form
The field of the -adic numbers is the completion of the field with respect to its -adic valuation (http://planetmath.org/PAdicValuation); thus may be thought the subfield (prime subfield) of . We can call the elements of the proper -adic numbers.
If, e.g., , we have the 2-adic or, according to G. W. Leibniz, dyadic numbers, for which every is 0 or 1. In this case we can write the sum expression for 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
(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
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 such that is the quotient field of this ring.
Every such 2-adic integer whose last bit is “1”, as , is a unit of this ring, because the division 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 , the generator of which may be taken (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 . The only non-trivial ideals in the ring of 2-adic integers are They have only 0 as common element.
All 2-adic non-zero integers are of the form where . The values 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 , then the 2-adic valuation of the 2-adic number is given by
|Title||p-adic canonical form|
|Date of creation||2013-03-22 14:13:37|
|Last modified on||2013-03-22 14:13:37|
|Last modified by||pahio (2872)|
|Defines||proper p-adic number|
|Defines||2-adic fractional number|