[FOM] Algebraic closure of Q

[Timothy Y. Chow]

Stephen G Simpson <simpson at math.psu.edu> wrote:
> Hodges' result is that, if you use Hodges' definition of algebraic
> closure, then ZF without the Axiom of Choice does not suffice to prove
> the existence of the algebraic closure of Q, the rationals.
      ^^^^^^^^^
"Existence" should be "uniqueness," of course.

I had a brief email discussion with Hodges, who corrected me, attributing 
this particular fact to Jech and Sochor and saying that his own results 
were related but different.  To set the record straight, let me quote from 
Hodges's paper:

H. Lauchli constructed, within a model of a weak form of set theory, an 
algebraic closure of the field Q of rationals which had no real-closed 
subfield.  Lauchli's construction is easily transferred to a model N of 
ZF, and it follows at once that neither of the two following statements is 
provable from ZF alone:

  (1) Every algebraic closure of Q has a real-closed subfield.
  (2) There is, up to isomorphism, at most one algebraic closure of Q.

(Cf. Jech and Sochor item (27).)  We shall show that none of the following 
statements is true in Lauchli's model:

  (3) There is a non-trivial absolute value v on L.
  (4) The Galois group Gal(L/Q) is non-trivial.
  (5) Every principal ideal domain has prime elements or is a field.

In particular, none of (3)-(5) is provable from ZF alone.

Refs:
H. Lauchli, Auswahlaxiom in der Algebra, Comment. Math. Helv. 37 (1962), 
1-18.
T. Jech and A. Sochor, Applications of the theta-model, Bull. Acad. Polon. 
Sci. Ser. Sci. Math. Astronom. Phys. 14 (1966), 351-355.

> My question is, does Hodges' result have any mathematical or
> foundational significance, beyond the above?  For example, is Hodges'
> model of ZF (used to prove his result) of additional interest in
> algebra, or perhaps in combinatorics, or in some other branch of
> mathematics?

The additional information I just provided about Lauchli's model may 
partially answer the question about what significance it has.  It seems to 
be useful for proving a number of independence results.  Whether it has 
interest in algebra or combinatorics, I don't know.  I get the impression 
that things provable in ZFC but not in ZF usually seem to be of interest 
only to those who are interested in foundations.  Hodges uses some fairly 
sophisticated number theory (e.g., class field theory) in his paper, but I 
would be surprised if one could turn things around and get applications to 
number theory from these results.

Tim