Re: Partitions of Reals

cbrown_at_cbrownsystems.com
Date: 02/11/05 

Date: 11 Feb 2005 10:26:03 -0800

Robert Israel wrote:

<snip proof that u = a - sqrt(b) in A if 1 in A>
>
>
> Question: are all positive algebraic numbers in A?
>

This doesn't answer your question, but it led me in this direction:

Let P be the set of all polynomials over Q with positive coefficients
and 0 constant coefficient.

Then, if a is any number in A, then surely for all p in P, p(a) is also
in A (this is just the closure of a and Q under addition and
multiplication). I'll write P(a) for {x : exists p in P s.t. x = p(a)}.

Then b must also be in A if there exists p, q in P such that p(a) =
q(b).

So there's a sort of minimal "ideal" which we could define around any
element a:

I(a) = {b : exists p,q in P s.t. p(a) = q(b)}

P(1) = Q, and I(1) contains all positive, real roots of polynomials
over Q with positive coefficients except for the constant coefficient,
which would be (strictly) negative. That's "a lot" of the algebraics,
but not all (not to imply that your conjecture is false...)

Cheers - Chas

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