Question on algebraic numbers

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? The fact that one cannot write any
algebraic formula for the roots of polynomials over Q of degree
greater than or equal to five does not mean that such roots cannot be
written as some finite expressions of integers. (To put that
differently, the insolvability of a polynomial over Q does not mean
that its roots may not happen to be expressible as finite expressions
involving integers that bear no generalizable algebraic relation to
the coefficients of the polynomial.)

In other words, are there algebraic numbers that not only cannot be
written as general finite expressions in the coefficients of
polynomials that may define them, but that cannot be written at all as
finite expressions involving integers, the arithmetic operations, and
raising to rational powers?
.

Relevant Pages

  • Re: Question on algebraic numbers
    ... In principle, that's really just the first step in the construction, ... adjoining to F1 all the roots of polynomials with coefficients in F1, ... written as some finite expressions of integers. ...
    (sci.math)
  • Re: Question on algebraic numbers
    ... adjoining to Q the roots of all polynomials over Q. ... written as some finite expressions of integers. ... involving integers (the algebraic relations to the coefficients of the ...
    (sci.math)
  • Re: Question on algebraic numbers
    ... adjoining to Q the roots of all polynomials over Q. ... written as some finite expressions of integers. ...
    (sci.math)
  • Re: MINIMAL POLYNOMIALS FOR COSINES
    ... polynomial constructed from these roots has rational ... over the rationals would general be twice phi. ... and cyclotomic polynomials. ...
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  • Re: how to construct
    ... and the rest of the roots are difficult to look al. ... It's not clear what you mean by "represent numbers as polynomials". ... all roots of f+ f- fare rationals ... general solution is f= n f, where fis an arbitrary polynomial ...
    (sci.math)
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