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?