[FOM] Query on p-adic numbers
joeshipman@aol.com
joeshipman at aol.com
Tue Apr 25 01:50:47 EDT 2006
Fix a prime number p, and let Qp be the field of p-adic numbers. There
is a subfield of Qp consisting of the "p-adic algebraic numbers", which
I will denote by Qp_a, which is elementarily equivalent to Qp and which
consists of all p-adic numbers which satisfy a polynomial equation over
Q.
Is there a "nice" enumeration of Qp_a?
My definition of "nice" is "nicer than the standard enumeration of the
algebraic numbers given by enumerating integer polynomials, casting out
the reducible ones, and ordering the roots of the irreducible ones
lexicographically by their real and imaginary parts". (We can obviously
do the same kind of thing for the p-adics since it is decidable which
polynomials have roots in the p-adics.)
I leave "nicer than" as an undefined term, but expect that there will
be much more of a consensus about statements of the form "X is nicer
than Y" than statements of the form "X is nice".
The reason I am not completely pessimistic about this is that the
arithmetic operations in p-adic fields have better algorithmic
convergence properties than they do in the real and complex number
fields.
-- JS
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