Re: Is it possible to generalise the Galois group definition for certain sets of polynomials
- From: Gerry <GerryMrt@xxxxxxxxx>
- Date: Fri, 28 Dec 2007 08:28:27 -0800 (PST)
On Dec 28, 4:38 pm, Chip Eastham <hardm...@xxxxxxxxx> wrote:
On Dec 28, 10:06 am, Gerry <Gerry...@xxxxxxxxx> wrote:
Hi all,
is it possible to generalise the Galois group definition for the
following sets of polynomials
P(x)=x^(6+4*n)+c*x^(2+4*n)-1
and
Q(x)=P(x)+2
n >= 0
c any complex number ;-)
with roots pi = ai + bi*I for polynomial P(x)
and roots qi = bi + ai*I for polynomial Q(x)
i=1 .. 4*n+6 the index for the roots
I=sqrt(-1)
Earlier this year there was a post on this subject under the
description:
"Transformation of complex arguments in polynomials" (June 22 Jürgen
Will)
I'm not sure how far they got but coincidentally i noticed some
polynomials
with this property and hence this post.
As an example:
For n=0, c=5 we get
P(x)=x^6+5*x^2-1
Q(x)=x^6+5*x^2+1
For n=2,c=1/3 we get
P(x)=x^14+1/3*x^10-1
Q(x)=x^14+1/3*x^10+1
Any comments are welcome.
Gerry
I'm not sure what in the way of generalization you are
looking for. The Galois group depends on an extension
of fields. Often the base field includes coefficients
of a polynomial while the extension field includes all
the roots of the polynomial in a minimal fashion over
the base field, and the Galois group of the polynomial
is then the group of field automorphisms of the
extension field which leave the base field fixed.
One possible quirk suggested by your post is that you
include a coefficient c which can be complex. So it
might be that you are thinking of the base field
being something other than the rationals Q.
But your examples do not bear out that suggestion,
as you take rational or integer values for c.
BTW, as you have set them up, there is a simple
relationship between roots of P(x) and Q(x) in
that Q(x) = -P(ix) when c is a real number.
P(x)=x^(6+4*n)+c*x^(2+4*n)-1
P(ix)=-x^(6+4*n)-c*x^(2+4*n)-1
-P(ix) = P(x)+2 = Q(x)
regards, chip- Hide quoted text -
- Show quoted text -
Hi Chip,
understood lets keep c rational.
With 'generalization' i was thinking of a possible relationship
between n
and the Galois group of P(x).
Regards
Gerry
.
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