# [FOM] Real and imaginary parts of algebraic numbers

Franklin franklin.vp at gmail.com
Thu Aug 1 01:12:12 EDT 2013

If m and n are coprime and x is algebraic of degree n and y is of degree m
then x+y is of degree n+m. I think therefore, as long as we can produce two
coprime numbers <=n (and >1) we will have RAlg_n[i] larger than ALg_n. So
for n>2 we shouldn't get equality.

For n=1 Alg_n doesn't contain i.
For n=2 Alg_n doesn't contain \sqrt{2}+i\sqrt{3}. The minimal polynomial
has degree 4.
For n=3 Alg_n doesn't contain \sqrt[3]{2}+i\sqrt[3]{3}. The minimal
polynomial has degree 18.

________________________________________________________
Franklin Vera Pacheco (Frank cheValier on a Pc)
www.franklinvp.com
University of Toronto

On Tue, Jul 30, 2013 at 12:56 PM, <joeshipman at aol.com> wrote:

>
> (2) If Alg_n is the field generated by all roots of polynomials over the
> rationals of degree ≤n, and RAlg_n is the field generated by all REAL roots
> of polynomials over the rationals of degree ≤n, for which n does
> Alg_n=RAlg_n[i]?
>
>
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