On Tue, Sep 7, 2010 at 6:55 AM, pgdoyle <[email protected]> wrote:
> I tried using Sage to compute the Galois group of a polynomial of
> degree 12, using the following code:
>
> p= -98298717579910546875000000000000000 +
> 3609188835688142358398437500000000* x -
> 11933130583987134521484375000000* x^2 + 7547595439287155273437500000*
> x^3 + 15827546508535476562500000* x^4 - 36495365154407915625000*x^5 +
> 44425378937243025000*x^6 - 40947614367603750*x^7 + 44073937859625*x^8
> - 44467752600* x^9 + 20580390*x^10 - 5090*x^11 + x^12
> K=NumberField(p,'a')
> K.galois_group()
>
> That got me this error message:
>
> NotImplementedError: You must install the optional Kash package to use
> Kash from Sage.
>
> When I tried to install Kash, I got the error message, `Error
> extracting kash' (see below).
>
> I was able to determine that the Galois group of this polynomial is
> the symmetric group by factoring the polynomial mod 23 and mod
> 364717651 to show that the group contains an 11-cycle and a 2-cycle.
> The Mathematica program I wrote to find the magic modulus 364717651
> took hours to run. I'm sure I could have figured out how to write the
> same program in Sage, if I had needed to. But this brute force
> approach just barely worked in this case. Next time the polynomial
> might have higher degree. Or its Galois group might not be the
> symmetric group.
>
> So my question is, how should I have approached this problem? Should
> I have tried harder to install Kash?
My preference would be that you try harder to install Kash, and
(a) learn the Sage spkg system and
(b) submit a fix for the problem you ran into.
Then other Sage users will benefit. None of the other options for what you
might do below benefit the community of Sage users.
> Used some other free package?
I am aware of no open source programs for computing Galois groups of
large degree polynomials.
Note that though KASH is "free" (as in beer) it is closed source.
> Bought a Magma license?
> Learned more Galois theory?
Yes, that would be optimal. It would be great if Sage (or some open
source package that Sage includes) could compute Galois groups of
larger degree polynomials... at least in some cases.
>
> Cheers, Peter
>
>
>
> ****************************************************
> Host system
> uname -a:
> Darwin gnumath.local 10.4.0 Darwin Kernel Version 10.4.0: Fri Apr 23
> 18:28:53 PDT 2010; root:xnu-1504.7.4~1/RELEASE_I386 i386
> ****************************************************
> ****************************************************
> CC Version
> gcc -v
> Using built-in specs.
> Target: i686-apple-darwin9
> Configured with: /var/tmp/gcc/gcc-5465~16/src/configure --disable-
> checking -enable-werror --prefix=/usr --mandir=/share/man --enable-
> languages=c,objc,c++,obj-c++ --program-transform-name=/^[cg][^.-]*$/s/
> $/-4.0/ --with-gxx-include-dir=/include/c++/4.0.0 --with-slibdir=/usr/
> lib --build=i686-apple-darwin9 --with-arch=apple --with-tune=generic --
> host=i686-apple-darwin9 --target=i686-apple-darwin9
> Thread model: posix
> gcc version 4.0.1 (Apple Inc. build 5465)
> ****************************************************
> Usage:
> List: tar -tf <archive-filename>
> Extract: tar -xf <archive-filename>
> Create: tar -cf <archive-filename> [filenames...]
> Help: tar --help
> Error extracting kash.
>
> real 0m0.040s
> user 0m0.031s
> sys 0m0.009s
>
> --
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>
--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org
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