Let
be an Euclidean domain with a regular
Euclidean size
.
Let
be a prime element.
Let
and let
such that
-
,
-
,
-
.
If
is unknown, but
and
are known
then we can compute a
-adic expansion of
as follows.
Let
be a
-adic expansion of
w.r.t.
.
Let
be a positive integer.
Recall that
the element
![$\displaystyle a^{(k)} = a_0 + a_1 p + \cdots a_{k-1} p^{k-1}$](img28.png) |
(1) |
is a
-adic approximation of
at order
.
Let us denote by
the canonical homomorphism from
to
.
The following formula computes
from
:
![$\displaystyle {\Phi}_p(\frac{{\phi}(a^{(k)})}{p^k}) = - {\Phi}_p({\phi}'(a_0)) {\Phi}_p(a_k)$](img34.png) |
(2) |
Marc Moreno Maza
2008-01-31