KASH  2.2
                   (version KANT V4, 10/99)

This is release 2.2 of KASH, the KAnt V4 SHell.

KANT V4 is developed by a research group at the Technische
Universit\"at Berlin under the project leadership of
Prof. Dr. M.E. Pohst. Its name is the abbreviation of

     Computational
     Algebraic
     Number
     Theory

with a slight hint of its german origin. As the name indicates, KANT
is a software package for mathematicians interested in algebraic
number theory. For those KANT is a tool for sophisticated computations
in number fields. With KASH you are able to use the powerful KANT V4
functions within a shell and you do not need to know anything at all
about programming in C.


KASH is freely available by ftp from

      ftp.math.tu-berlin.de

    where it sits in the subdirectory /pub/algebra/Kant/Kash

    or

    ftp://ftp.math.tu-berlin.de/pub/algebra/Kant/Kash


/***** NEW FEATURES ** NEW FEATURES ** NEW FEATURES ** NEW FEATURES **********/

o  divisor class groups of global function fields
o  divisor class representation
o  divisor reduction, extended Riemann-Roch space computations
o  S-units for global function fields
o  differentials, canonical divisors, differential spaces, differentiations
o  Cartier operator for global function fields
o  Hasse-Witt invariant of a global function field
o  L-polynomial computation for global function fields
o  gap numbers, Weierstrass places
o  other functions for global function fields

o  improved class group, ray class group and S-unit computation 
o  optimized power products
o  support functions for class fields


/******* INSTALLATION ** INSTALLATION ** INSTALLATION ** INSTALLATION ********/

To make your life easier we provide binaries of the shell. At the
moment we are supporting the following architectures:

o  DEC  alpha   : Digital Unix 4.0
o  HP   PA-RISC : HP-UX 10.20
o  IBM  RS6000  : AIX 4.2
o  SUN  SPARC   : SunOS 4.1.3
o  SUN  SPARC   : SunOS 5.7
o  SGI  MIPS    : Irix 6.5
o  PC   i486    : Linux 2.2.34 (elf)
o  PC   i486    : IBM OS/2 Warp 3.0
o  PC   i486    : MS DOS  5.0
o  PC   i486    : MS Windows 3.1
o  PC   i486    : MS Windows 95
o  PC   i486    : MS Windows NT

For all of the above versions you'll have to get TWO files,
Kash_2.2.***.tar.gz    (for the binary)
and
Kash_2.2.common.tar.gz (containing the library, documentation, ...)

o  PC  (80{4,5,6}86)  : IBM OS/2 4.0 
o  PC  (80{4,5,6}86)  : MS DOS  5.0   
o  PC  (80{4,5,6}86)  : MS Windows 95 
o  PC  (80{4,5,6}86)  : MS Windows NT

All files needed for the DOS/ OS/2/ Windows version are located in a
subdirectory
Kash_2.2.4.OS2_DOS_Windows
Here you'll find installation guides for this version.


/**** KASH 2.2 ** KASH 2.2 ** KASH 2.2 ** KASH 2.2 ** KASH 2.2 ** KASH 2.2 ***/

The main features of the current release are (plus the new features mentioned
above):

Computations in number fields

o  arithmetic of algebraic numbers,
o  computation of maximal orders in number fields,
o  modular computation of resultants,
o  unconditional and conditional (GRH) computation of class groups
   of number fields,
o  unconditional and conditional (GRH) computation of fundamental
   units in arbitrary orders,
o  S-unit computation,
o  computation of all subfields of a number field,
o  determination of Galois groups of number fields up to degree 15,
o  ray class groups, 
o  automorphisms of normal and abelian fields,


Ideals in number fields

o  arithmetic of fractional ideals in number fields,
o  computation of prime ideal decompositions of fractional ideals
   in number fields,
o  (ray) class group representation of an ideal (discrete 
   logarithm for ray class groups),
o  computation of the multiplicative group of residue rings of maximal 
   orders modulo ideals and infinite primes, 
o  Chinese remainder for ideals and infinite places,


Relative extensions of number fields

o  computation of maximal orders (relative Round 2),
o  arithmetic of algebraic numbers,
o  signature of polynomials,
o  normal forms of modules in relative extensions,
o  arithmetic of relative ideals,
o  computation of a 2-element-representation for relative ideals,
o  Kummer extensions of prime degree, relative field discriminant and
   integral basis,

Computations in class field theory

o  Hilbert and ray class fields of imaginary quadratic fields by
   complex multiplication analytic methods,
o  Hilbert class fields of totally real number fields via the 
   computation of Stark units by Stark's conjecture,
o  computation of discriminants and conductors of ray class fields,
o  computation of ray class fields, subgroups of ray class
   groups supported,
o  Artin map and automorphisms of ray class fields,

Galois groups

o  computation of Galois groups of polynomials over Q
   up to degree 15 and their representation as permutation
   groups on the roots,
o  symbolic computation of Galois groups of polynomials over Q,
   number fields and Q(x) up to degree 7

Lattices

o  lattices and enumeration of lattice points,
   lattices and lattice reduction for lattices over number fields,


Diophantine equations

o  Thue equation solver,
o  unit equations and exceptional units,
o  index form equations,
o  integral points on Mordell curves,
o  norm equation solver for absolute and relative extensions,

Algebraic function fields over finite fields, Q or number fields

o  arithmetic of algebraic functions,
o  genus computation,
o  places, divisors and Riemann-Roch spaces,
o  dimension of exact constant field,
o  computation of maximal orders in function fields,
o  arithmetic of fractional ideals of orders of function fields,
o  computation of prime ideal decompositions of fractional ideals,
o  basis reduction, fundamental unit computation in global
   function fields,
o  determination of places of degree one in global function fields,
o  divisor class group (of degree 0) for global function fields
o  divisor reduction, divisor class representation
o  S-units for global function fields
o  differentials, differential spaces, differentiations
o  Cartier operator for global function fields
o  Hasse-Witt invariant of a global function field
o  L-polynomial computation for global function fields
o  gap numbers, Weierstrass places


Specials

o  factorization of polynomials over number fields,
o  basic linear algebra over number fields,
o  a package for Abelian groups, 



/****** SUPPORT ** SUPPORT ** SUPPORT ** SUPPORT ** SUPPORT ** SUPPORT *******/


Please mail all your questions, suggestions, comments and bug reports
concerning KASH to

   kant@math.tu-berlin.de



/******** ACKNOWLEDGMENTS ** ACKNOWLEDGMENTS ** ACKNOWLEDGMENTS *************/


We would like to thank

o   Prof. J. Cannon at the University of Sydney, for the opportunity
    of using the MAGMA C-kernel for the development of KANT V4,
    the algorithmic part of KASH.

    Special thanks to Wieb Bosma and Allan Steel for their help.
    It would have been impossible to develop this software without
    their help.


o   Prof. Dr. J. Neub\"user at the RWTH Aachen, F.R.G.,
    for his permission to use and modify large parts of the GAP source code.
    Especially, we would like to thank M. Sch\"onert, who mainly created
    GAP, for his kind support and help.


o   Dr. A. Weber at Cornell University, USA, for his work on the database.


o   Dr. M. Klebel at Augsburg for his work on class fields
    of imaginary quadratic number fields.


o   Dr. A. Hulpke at St. Andrews for his support for the Galois package.