[FOM] Galois theory in EFA, for FLT
Colin McLarty
colin.mclarty at case.edu
Fri Nov 15 09:34:21 EST 2013
A point about EFA and a request for references if they exist:
The first order theory of a field K can code the Galois group of any finite
extension of K. Indeed it can encode the Galois groups of all degree n (or
smaller) extensions of K uniformly in the coefficients of a degree n
polynomial defining the extension -- but not uniformly in the degree. (For
exposition see Chatzidakis "On the Cohomological Dimension of Non-standard
Number Fields".) So you cannot code quantification over arbitrary degree
extensions.
Number theorists often use the absolute Galois group of a field K, which is
uncountable and not interpretable in the first order theory of K. Part of
this goes easily into PA since they typically use continuous
representations of K in finite fields F and these amount to representing
Galois groups of finite extensions of K (in those same fields F). But the
absolute Galois group also gives the connections among all finite
extensions, and only limited aspects of that are expressible in the first
order theory. The first order theory cannot express the general idea of
"for each degree n extension there exists a possibly higher degree
extension such that...." The absolute Galois group can -- and that is
often the point of using the absolute Galois group.
My point is that, because Peano Arithmetic (PA) and its fragment Elementary
Function Arithmetic (EFA) are sequential theories, they can code
polynomials uniformly in the coefficients and degree and even the finite
number of variables. Then the construction of Galois groups is so
uncomplicated that EFA can quantify uniformly over the Galois groups of all
finite extensions of the rationals. So EFA can interpret much more of the
theory of the absolute Galois group of the rationals than the first order
theory of a field K can about that of K.
My question is: can anyone give me a citeable reference for using either PA
or any fragment to code the Galois groups of arbitrary finite extension of
Q, uniformly so PA can quantify over these groups?
best, Colin
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