[FOM] Real and imaginary parts of algebraic numbers
franklin.vp at gmail.com
Thu Aug 1 00:43:52 EDT 2013
If r is a root of P(x) and s is a root of Q(x) then x+y is a root of
the resultant of P(x) and Q(z-x) considering Q(z+x) a polynomial in x,
and rs is a root of the resultant of P(x) and x^deg(Q)Q(z/x). The
resultant vanishes iff P(x) and Q(z-x) have a common root. Notice that
if z=r+s then P(r)=0 and Q((r+s)-r)=Q(s)=0. Likewise for P(x) and
x^deg(Q)Q(rs/x). The resultant has degree equal deg(P)deg(Q). We have
that a+ib is root of a polynomial of degree n, a-ib is too, the same
polynomial. Then 2a=a+ib+a-ib is a root of a polynomial of degree 2n,
and therefore a is too. Therefore a bound for a is 2n.
Likewise ib is root of a polynomial of degree 2n and i is root of a
polynomial of degree 2, x^2+1. So, from the above we get that b is a
root of a polynomial of degree at most 4n.
Franklin Vera Pacheco (Frank cheValier on a Pc)
University of Toronto
On Tue, Jul 30, 2013 at 12:56 PM, <joeshipman at aol.com> wrote:
> (1) If x=a+bi is a root of an irreducible polynomial over the rationals of
> degree n, what is the maximum possible degree of the irreducible polynomials
> over the rationals for the real numbers a and b?
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