[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 

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 

-- JS

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