From: Torsten Ekedahl
Subject: Re: Galois - A new question? How close?
Date: 30 Mar 1999 06:58:30 +0200
Newsgroups: sci.math,sci.math.research
Keywords: Galois groups for nearby polynomials
mckay@cs.concordia.ca (MCKAY john) writes:
> Given f in Q[x], and an appropriate norm, || . ||, what can one
> say of the galois group/Q of polynomials arbitrarily close to a
> given polynomial?
>
> An example: Let f = x^7-7x+3 (Trinks' polynomial), Gal(f)/Q = PSL(3,2)
> of order 168.
> What is known of Gal(g)/Q where g = y^7-ay+b and |7-a|+|3-b| < epsilon?
>
If there is any choice of rational a and b for which Gal(g) is S_7
then there is one for which |7-a|+|3-b| < epsilon. This is done by
choosing primes p with Frobeniuses (for the given choice of a and b)
having conjugacy classes in S_7 such that any choice of elements in
those conjugacy classes generate S_7 and then choosing a' and b' close
in real absolute value to 7 and 3 and close p-adically for those
primes to a and b. This is possible by the strong (even weak I guess)
approximation theorem.
On the other hand if for all rational a and b Gal(g) is smaller than
S_7 then by the Hilbert irreducibility theorem Gal(g) where a and b
are polynomial variables Gal(g)/Q(a,b) would also be smaller than S_7
which can be shown to be false in many ways.
One way to do that (and to directly give rational a and b as well) is
to note that if g is irreducible then it is enough to find one prime
for which the reduction of g has five distinct linear and one square factor as
then the Frobenius will be a transposition and it together with the
7-cycle that is forced by irreducibility generate s_7. For this one
can take p=257, a=2, and b=258. Hence as soon as
|a'-2|_257, |b'-258|_257 < 1
and y^7-a'*x+b' is irreducible Gal(g)=s_7.
==============================================================================
From: Kurt Foster
Subject: Re: Galois - A new question? How close?
Date: 30 Mar 1999 16:55:24 GMT
Newsgroups: sci.math,sci.math.research
In sci.math MCKAY john wrote:
. Given f in Q[x], and an appropriate norm, || . ||, what can one
. say of the galois group/Q of polynomials arbitrarily close to a
. given polynomial?
. An example: Let f = x^7-7x+3 (Trinks' polynomial), Gal(f)/Q = PSL(3,2)
. of order 168.
. What is known of Gal(g)/Q where g = y^7-ay+b and |7-a|+|3-b| < epsilon?
Over Q? Not much. The discriminant being the square of a rational
number, is one obvious property pertinent to the structure of the Galois
group, that is not likely to persist throughout an open neighborhood in Q,
of a set of coefficients for which the discriminant is a perfect square.
Over completions of Q at nonarchimedean valuations, though, it's a VERY
different story. See "Krasner's Lemma" in any of a number of books on
algebraic number theory, and the more detailed sequela in the literature.
If memory serves, Krasner wrote a long paper detailing just how close was
"close enough" in this context.
The Galois groups of trinomials x^n - ax + b over Q are "usually" the
whole symmetric group S_n, aren't they?