Re: The algebraic closure of the rationals
- From: "joeshipman@xxxxxxx" <JoeShipman@xxxxxxx>
- Date: 7 Apr 2006 06:12:32 -0700
There is a monograph by Brawley and Schnibben ("Infinite algebraic
extensions of finite fields", AMS Contemporary Mathematics series)
which treats this. In fact, it's not hard to get an "explicit
description" of any algebraic extension of a finite field -- they are
characterized by the orders of their finite subfields, which in turn
are characterized by functions from the set of primes to
{0,1,2,3,...,"infinity"} -- for any such function f, the corresponding
algebraic extension of Z/p has a finite subfield of order p^k iff the
prime factorization of k has the exponent of each prime q being <=
f(q).
If you are just interested in algebraic closures of finite fields,
there is an especially nice and explicit representation due to Conway
in the case of characteristic 2. The ordinal omega^(omega^omega) forms
an algebraically closed field of characteristic 2 under "nim-addition"
and "nim-multiplication". See Conway's book "On Numbers and Games", or
H.W. Lenstra's paper "Nim Multiplication" which can be found here:
http://hdl.handle.net/1887/2125
It is an interesting open problem to extend this construction to
characteristics other than 2. The idea is to define addition and
multiplication as far as possible inductively, and when you have a
choice pick the least ordinal consistent with the resulting structure
being part of a field; but in characteristic p>2 it is hard to find a
nice interpretation of the operations that makes them easy to compute.
-- Joe Shipman
tchow@xxxxxxxxxxxxx wrote:
3) I've heard that it's even hard to get an "explicit" description
of the algebraic closures of finite fields - are there any
theorems to this effect?
I doubt it. Probably what underlies the rumor is the fact that if you
want to construct F_{p^n} then you need to pick an irreducible degree-n
polynomial over F_p, and in general there will be many choices, none of
which will be "canonical" or "natural." However, as far as computability
or constructibility goes, you can always factor x^(p^n) - x and make up
some rule for picking a suitable irreducible polynomial---say, the first
one lexicographically or something. The list of polynomials you get as
you go up to higher and higher degrees will not have a "simple explicit
description," but I doubt there is a satisfactory way to define the word
"explicit" precisely enough to let you prove theorems about it.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
.
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