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? Used some other free package? Bought a Magma license? Learned more Galois theory? 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 -- To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
