COMPUTATION OF THE GALOIS GROUP OF A POLYNOMIAL WITH DEGREE < 12 Copyright M. Olivier and Y. Eichenlaub (olivier@math.u-bordeaux.fr) Version 1.5 Last released: 1996, April 01. The directory contains 3 files and a lot of data : galp is an executable file for SUN/SPARC station 1,2,4,5,10,20,... galp.c is a source file ; the compilation requiests the installation of the PARI library makefile is an example of makefile to compile galp.c All the others files are binary data files : RESxxx are the invariant polynomials for degrees 8, 9, 10 and 11 COSxxx are the relative cosets of the transitive groups for degrees 8, 9, 10 and 11 (For degree <= 7, the method used is the one of PARI/GP package) Example 1 : compute the Galois group of x^8-2 Invoke galp Type x^8-2 or poly([1,0,0,0,0,0,0,0,-2],x) The answer is T_8[8] which means the transitive group of degree 8 numbered 8 in the paper : "The transitive groups of degree up to eleven" by G. Butler and J. McKay, in Communications in Algebra, vol. 11, 1983, pages 863--911. Example 2 : compute the Galois group of x^5-2 Type x^5-2 or poly([1,0,0,0,0,-1],x) The answer is M_20 the metaplectic group of order 20 It is possible to invoke galp with at most two input parameters: The first one is a flag (0 or 1) which indicates that you want that galp verboses the execution of the program; The second one is the precision of the initial computations, i.e. an integer n which is the number of figures for the computations. The syntax will be: galp flag precision In case of a polynomial with large coefficients, it is better to invoke galp with a given precision (and consequently also a flag value). The defaults are flag=0 and precision=28 decimal digits. Note that the polynomial has to be irreducible and with coefficients in Z.