# [FOM] Algebraic closure of Q

joeshipman@aol.com joeshipman at aol.com
Tue May 16 15:10:19 EDT 2006

```We have been discussing how much choice is needed to prove the
existence and uniqueness of the algebraic closure of Q. But from a
constructive point of view, what matters is that we have a way to
enumerate this field and compute the field operations.

(Even if some choice is required to prove that the construction has
certain properties, the construction itself is just a set of algorithms
to enumerate elements of Qbar and calculate the field operations.)

What I would like to know is how "nice" such a construction can be.
The one I am thinking of represents algebraic numbers by their
irreducible polynomials over Z, distinguishing between conjugates using
the lexicographic ordering (the "first root" is the one which has the
smallest imaginary part among those roots with the smallest real part,
etc.).  The algebra is messy and painful to implement, but
straightforward.

But much nicer representations exist for other fields. For example,
Conway has found natural and nicely computable operations on ordinals,
"nim-addition" and "nim-multiplication", under which the ordinal
omega^(omega^omega) is the algebraic closure of the field 2={0,1}.
Explicit constructions for the algebraic closure of other finite fields
have been given by Brawley and Schnibben (though finding a construction
for each characteristic that is as "canonical" as Conway's remains
open).

The reason these representations are "nicer" is that the representation
of the elements is straightforward -- it is immediately obvious whether
a particular term in the notation system represents a field element,
and notations are unique, so that it is easy to calculate an
enumeration (given n, we can quickly write down the element of the
field; this is especially easy in characteristic 2, because Conway's
construction means one merely has to enumerate omega^(omega^omega) ).

In the representation of algebraic numbers I describe above, it is easy
to recognize that you have a pair (integer polynomial of degree n,
integer in {1,...,n}), but it is not easy to confirm that such a
notation actually represents a field element, because the polynomial
must be tested for irreducibility. (Also, notations are not unique, but
that's easier to fix, because we can require that the polynomial have
no common factor for the coefficients and positive leading term, which
are easily checked.)

This means that any conversion of the representation to an enumeration
will be really slow (to find the nth algebraic number you have to
enumerate polynomials and sieve out the reducible ones, unless there is
a way of enumerating irreducible polynomials only that I don't know

Another field I'd like to see a much better representation for is the
"p-adic algebraic numbers" -- given p, this is the smallest field
elementarily equivalent to the p-adics. Although the best known
algorithm for the first-order theory of the p-adic numbers is
non-elementary, the p-adic algebraics ought to have a reasonable
representation since it is much easier than that to decide which
polynomials have roots in the p-adics (though I don't know precisely
how easy). Once you have irreducible polynomials for algebraic p-adic
numbers (plus a "which root" flag), the computation of the field
operations is even easier than it is for the ordinary algebraic numbers
viewed as a subfield of the complex numbers.

(Whether there is a nice operation-preserving mapping from the p-adic
algebraics into the ordinary algebraics seems a much more difficult
problem -- it's easy to find conjugates, but hard to consistently
select one of them.)

-- JS
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