Computing the Galois group of a degree 12 polynomial
pgdoyle
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? Used some other free package?
Bought a Magma license? Learned more Galois theory?
Cheers, Peter
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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
William Stein
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.
> --
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--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org