On 25 July 2012 00:30, Justin C. Walker <[email protected]> wrote: > While looking through Stein's "Three Lectures" paper, I tried the examples > in \S 2.1.1. In particular, the last item, computing the order of the cubic > field's Galois group (72) seems to be straightforward when looking at the > paper. When I tried this in Sage 5.1, the computation twiddles away for > roughly 20 minutes, and then blows up, complaining that "you must specify > the name of the generator." > > Has this changed since the paper was written?
The problem with computing Galois groups is that it depends on what you mean by "compute the Galois group of f". The easy question is to determine the isomorphism class of the Galois group of f as an abstract permutation group. The hard question is to compute a defining polynomial for the splitting field of f, identify all the roots of f as elements of that field, and determine each element of the Galois group in terms of how it acts on a single generator of the splitting field. At some point in the past when William gave his Bordeaux lectures, Sage could only do the former (by wrapping the corresponding PARI command), which has the advantage of being very quick. Subsequently, I added functionality to do the latter, and this was made the default for the "galois_group" method of number fields (but *not* of bare polynomials). This is much much slower, and also occasionally prompts the user for a variable name for the splitting field when this is bigger than the given number field (i.e. when the given field isn't Galois). So I suspect that's what has mangled William's example. David -- -- To post to this group, send an email to [email protected] To unsubscribe from this group, send an email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
