  This directory contains some subdirectories with examples of SIMATH
main programs. Some of these programs are large and not easy to
understand, especially if you don't have the required mathematical
background. We distribute these files in order to help you write more
complex applications in SIMATH. You may find solutions to some higher
computational problems in the subdirectories.

  This is a short description of the programs:



  Groebner_basis/m_groebner_gfs.S:
	Groebner basis algorithm over a Galois-field.

  Groebner_basis/m_groebner_i.S:
	Groebner basis algorithm over the integers.

  Groebner_basis/m_groebner_mip.S:
	Groebner basis algorithm over Z/pZ where p is an arbitrary
	prime. (If p is < BASIS (BASIS = 2^30 = 1073741824), you should
	use the msp-program.)

  Groebner_basis/m_groebner_msp.S:
	Groebner basis algorithm over Z/pZ where p is a prime < BASIS,
	BASIS = 2^30 = 1073741824.

  Groebner_basis/m_groebner_nf.S:
	Groebner basis algorithm over a number field.

  Groebner_basis/m_groebner_pi.S:
	Groebner basis algorithm over polynomials over the integers.

  Groebner_basis/m_groebner_r.S:
	Groebner basis algorithm over the rationals.
	
  Groebner_basis/m_groebner_rfr.S:
	Groebner basis over rational functions over the rationals.



  function_fields/ORDMAXff.S:
	Let Z/pZ (X) denote the rational congruence function field of
	characteristic p. For a given monic separable polynomial F(Y)
	in Z/pZ (X) [Y], the program computes an integral basis, i.e. a
	Z/pZ[X] - basis of the ring of integers O of the algebraic
	congruence function field Z/pZ(X) (Y) / (F(Y) * Z/pZ(X) (Y)).

  function_fields/DECLAWff.S:
	Let Z/pZ (X) denote the rational congruence function field of
	characteristic p. For a given monic separable polynomial F(Y)
	in Z/pZ (X) [Y] and a given monic irreducible polynomial P(X)
	in Z/pZ [X], the program computes ramification indices and
	related residue class degrees of P in the ring of integers O of
	the algebraic congruence function field 
	Z/pZ(X) (Y) / (F(Y) * Z/pZ(X) (Y)).



  number_fields/ORDMAXnf.S:
	For a given monic irreducible polynomial F(x) in Z[x], the
	program computes an integral basis, i.e. a Z-basis of the ring
	of integers O of the number field Q[z] := Q(x) / (F(x) * Q(x)),
	and the index [O : Z[z]].

  number_fields/DECLAWnf.S:
	For a given monic irreducible polynomial F(x) in Z[x] and a
	given prime number p in Z, the program computes ramification
	indices and related residue class degrees of p in the ring of
	integers O of the number field Q[z] := Q(x) / (F(x) * Q(x)).
