On Mon, Feb 15, 2010 at 3:34 PM, Henry Baker <hbaker1@pipeline.com> wrote:
This may be what you want: "real closed fields" encompass all of the solutions of polynomials with integer coefficients & is closed under addition, subtraction, multiplication & division & roots & powers (so long as the roots & powers have fixed constant integer roots & powers). [I believe that the second half of the previous sentence is redundant.]
That's not what he wants. A real closed field is an ordered field in which every odd-degree polynomial has a root. It doesn't encompass all the solutions of polynomials: x^2 + 1 = 0 cannot have root in a real closed field; the ordering axioms make it impossible. But Dan wants only solutions of polynomials that are solvable in radicals. So a real closed field must have an x with x^5 + x = 1, but Dan's field does not.
Clearly the elements of real closed fields are a countable set.
Not true. Any field that satisfies the axioms of a real closed field is a real closed field. The smallest such is countable, but there are uncountable ones too, like the reals. From a logician's point of view, the interesting thing about the real-closed-field axioms is that they are categorical; any statement you can make in a language which talks about elements, +, and *, but not sets of elements, is either true of all real closed fields or false for all of them. Andy