A p-adic number is called unit if it is not a multiple of a negativepower of p
and its first digit is
nonzero. For example, and
are units while
not. A non-unit p-adic
number α can always be written in the form
where γ is a unit. For example,
, then .
We can thus restrict multiplication ofany two
p-adic numbers to multiplication of units. The multiplication can then be
carried similar to the
case of p-ary numbers. To multiply 2/3 and 5/6, we get the Hensel codes‘
The multiplication operation is illustrated below:
Thus, the result is
which is equal to
Again, we will only consider the division of p -adic units. Consider the
following p-adic units:
with . The quotient α
= δ/β can be written
where are the digits
of α. Since δ = β · α, we have
Even though the p-adic digits
in the interval [0, p − 1], we cannot assume that the
integers lie in this interval. Hence we
where ∈ [0, p − 1].
Then is the first digit in the p-adic
expansion for βα and is the carry
which must be added to . Thus,
This turns out to be the rule for obtaining each digit of
the expansion for α. At each stage of
the standard long division procedure, we multiply
(mod p) by the first digit of the partial
remainder and reduce the result modulo p.
As an example, we divide 2/3 by 1/12. We have
The first digit of the divisor is
and its multiplicative inverse modulo 5 is
The first digit of the partial remainder (which, in the
first step , is the dividend) is , which
This procedure produced the partial remainder which is
zero , hence we terminate the expansion.
In general, this will not happen and we will have to continue until the period
is exhibited. As a
check we observe that 2/3 ÷ 1/12 = 8 and 8 = .31 for p = 5.
We note that the division of p-adic numbers is deterministic and not subject to
trial and error
as is the case for division of p-ary numbers.
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