Re: Question on algebraic numbers

On Feb 1, 3:54 pm, dmr5...@xxxxxxxxx wrote:
The algebraic closure of Q (the rationals) is, of course, formed by
adjoining to Q the roots of all polynomials over Q.

Consider now the field consisting of all numbers that can be written
as finite expressions involving addition, subtraction, multiplication,
and division of integers, raising to rational powers, and compositions
of these operations.  This is obviously a subfield of the algebraic
closure of Q.

Question: is it a proper subfield?  

Yes, it is. What you have described is the maximal prosolvable
extension of Q. That is, it is the direct limit of all the finite
extensions of Q which have a solvable Galois group.

None of the roots of the polynomial x^5-4x+2 would be included in the
field you described. The Galois group of that polynomial over Q is
known to be S_5, which is not solvable.

.

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