On Monday, 28 April 2014 10:41:20 UTC+1, John Cremona wrote: > > Let's think about what users actually want when they call for the Galois > Group of a (possibly non-Galois) number field K. First of all, they might > want to know whether or not K is Galois itself; > this is basically comparing the degree of the defining polynomial with the order of its Galois group (cf. K.is_galois??), and can (and should - currently it is not) be done without constructing the Galois closure of K.
then they may want the Galois group of the closure > again, this does not need the closure, as the closure is just the splitting field of the defining polynomial (unless I mix things up...) The structure of the group is better investigated in a permutation representation of a smaller degree, that's clear, too. Dima > and the closure itself; about the group, they may only want to know > something about it such as its order, or its isomorphism type; and finally > they may want to use the elements of the group as automorphisms, i.e. as > maps from the Galois closure to itself. If there is a quicker way to get > the group order and/or structure without the rest, that should be available. > > John > > > On 28 April 2014 09:42, Dima Pasechnik <[email protected] <javascript:>>wrote: > >> On 2014-04-28, Rob Beezer <[email protected] <javascript:>> wrote: >> > The groups are isomorphic: >> > >> > sage: K.<a> = NumberField(x^4 - 2) >> > sage: G1 = K.galois_group(names='bbb') >> > sage: G2 = K.galois_group(type="gap",names='bbb').group() >> > sage: G1.is_isomorphic(G2) >> > True >> > >> > but >> > >> > sage: K.galois_closure(names="ccc") >> > Number Field in ccc with defining polynomial x^8 + 28*x^4 + 2500 >> > >> > So "bug" sounds overly harsh to me. Is GAP cleaning up it's version, by >> > replacing it with an isomorphic version? >> In case of G2, no attempt to work out the Galois closure is made. >> Sage basically returns the result of the following computation: >> >> sage: R.<x>=QQ[] >> sage: p = x^4 - 2 >> sage: p.galois_group() >> Transitive group number 3 of degree 4 >> >> This still sounds like a documentation bug to me, no? >> >> > >> > Pedagogically, I prefer G1, which uses 8 points, rather than the 4 used >> by >> > G2. >> >> computationally, G1 is often infeasible in cases where G2 is still >> quick to find, as Nils pointed out, too. >> >> Dima >> > >> > Rob >> > >> > On Sunday, April 27, 2014 3:32:46 PM UTC-7, Dima Pasechnik wrote: >> > >> >> Is this a bug? >> >> I ran into this while working on #16243. >> >> >> > >> >> -- >> You received this message because you are subscribed to the Google Groups >> "sage-devel" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected] <javascript:>. >> To post to this group, send email to [email protected]<javascript:> >> . >> Visit this group at http://groups.google.com/group/sage-devel. >> For more options, visit https://groups.google.com/d/optout. >> > > -- You received this message because you are subscribed to the Google Groups "sage-nt" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send an email to [email protected]. Visit this group at http://groups.google.com/group/sage-nt. For more options, visit https://groups.google.com/d/optout.
