KASH - Shell developed to make KANT easier to use

Version:KASH is a shell developed to make KANT easier to use. KANT is a software package for mathematicians interested in algebraic number theory. For those KANT is a tool for sophisticated computations in number fields, in global function fields, and in local fields.3

License:GPL

Operating System:Mac OS X

Homepage: www.math.tu-berlin.de

Developed by:

This shell is based on that of the group theory package GAP3 and the handling is similar to that of MAPLE. We put great effort into enabling the user to handle the number theoretical objects in the very same way as one would do using pencil and paper.

For example, there is just one command Factorization for the factorization of elements from a factorial monoid like rational integers in Z, polynomials over a field, or ideals from a Dedekind ring.

KANT consists of a C--library of thousands of functions for doing arithmetic in number fields, function fields, and local fields. Of course, the necessary auxiliaries from linear algebra over rings, especially lattices, are also included.

The set of these functions is based on the core of the computer algebra system MAGMA from which we adopt our storage management, base arithmetic, arithmetic for finite fields, polynomial, and linear algebra and a variety of other tools. In return, almost all KANT routines are included in MAGMA.

Kash is cross-platform and is available for Mac OS X, Linux/x86, and MS Windows.

Here are some key features of "KASH":

Computations in number fields

· arithmetic of algebraic numbers,

· computation of maximal orders in number fields,

· modular computation of resultants,

· unconditional and conditional (GRH) computation of class groups of number fields,

· unconditional and conditional (GRH) computation of fundamental units in arbitrary orders,

· S-unit computation,

· computation of all subfields of a number field,

· determination of Galois groups of number fields up to degree 15,

· ray class groups,

· automorphisms of normal and abelian fields,

Ideals in number fields

· arithmetic of fractional ideals in number fields,

· computation of prime ideal decompositions of fractional ideals in number fields,

· (ray) class group representation of an ideal (discrete logarithm for ray class groups),

· computation of the multiplicative group of residue rings of maximal orders modulo ideals and infinite primes,

· Chinese remainder for ideals and infinite places,

Relative extensions of number fields

· computation of maximal orders (relative Round 2),

· arithmetic of algebraic numbers,

· signature of polynomials,

· normal forms of modules in relative extensions,

· arithmetic of relative ideals,

· computation of a 2-element-representation for relative ideals,

· Kummer extensions of prime degree, relative field discriminant and integral basis,

Computations in class field theory

· Hilbert and ray class fields of imaginary quadratic fields by complex multiplication,

· Hilbert class fields of totally real number fields via the computation of Stark units by Stark's conjecture,

· computation of discriminants and conductors of ray class fields,

· computation of ray class fields, subgroups of ray class groups supported via Kummer theory

· Artin map and automorphisms of ray class fields,

Galois groups

· computation of Galois groups of polynomials over Q up to degree 21 and their representation as permutation groups on the roots,

· symbolic computation of Galois groups of polynomials over Q, number fields and Q(x) up to degree 7

Lattices

· lattices and enumeration of lattice points,

· lattices and lattice reduction for lattices over number fields,

Diophantine equations

· Thue equation solver,

· unit equations and exceptional units,

· index form equations,

· integral points on Mordell curves,

· norm equation solver for absolute and relative extensions,

Algebraic function fields over finite fields, Q or number fields

· absolute and relative extensions of algebraic function fields,

· arithmetic of algebraic functions,

· genus computation,

· places, divisors and Riemann-Roch spaces,

· dimension of exact constant field,

· computation of maximal orders in function fields,

· arithmetic of fractional ideals of orders of function fields,

· computation of prime ideal decompositions of fractional ideals,

· basis reduction for orders, fundamental unit computation in global function fields,

· determination of places of degree one in global function fields,

· S-units for global function fields

· differentials, differential spaces, differentiations

· Cartier operator for global function fields

· Hasse-Witt invariant of a global function field

· L-polynomial computation for global function fields

· gap numbers, Weierstrass places

Local rings and fields

· unramified extensions of local fields

· ramified extension of local fields

· factorization of polynomials over local fields,

Qaos Database Support

· access functions to the Qaos databases of number fields and transitive groups