order2
......
2008-10-06
1
1
experimental
http://www.math.tu-berlin.de/~kant/order2.ocd
arith1
fieldname1
matrix1
nums1
order1
polyd1
ring1
ringname1
setname1
This CD extends the order1 CD. It defines several basic functions for
orders of number fields and also states with the square matrices
constructions.
Written by Sylla Lesseni.
A CD of several basic functions for orders of number fields and
constructors of square matrices written for SCIEnce project.
Note that all the rings used here will be considered unital.
The reference textbooks about that are stated in this OM CD are:
(1) M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory,
Cambridge Univ. Press, 1989.
(2) H. Cohen, A course in Computational Algebraic number Theory.
Berlin, Springer-Verlag (1993).
element_of
application
This symbol represents a binary function. The first argument is any
non-empty set R and the second argument is a priori an element b of
a subset of R. When evaluated on R and b, the function returns the
element b coerced into R. For example this function could be used to
coerce an element of an order into
its maximal order.
1
1
2
2
0
1
1
3
3
0
OMI>5
basis
application
This unary function represents the constructor of a basis for an ideal
generated in an order (as a list of algebraic integers).
2
1
1
2
2
0
unit_rank
application
This unary function gives the rank of the unit group of the given
maximal order. Note that the argument could also be a number field.
1
1
2
-5
0
class_number
application
This symbol is a unary function; its argument is a maximal order
of a number field. It returns the class number of the given maximal
order. Note that the argument could also be a number field.
1
1
2
2
0
matrix_ground_ring
application
This symbol is a unary function, whose argument should be a ring R.
When applied to R, it represents the square matrix algebra ground
domain.
dimension
application
This symbol is a unary function whose argument must be a non-negative
OpenMath integer. When applied this creates an object that denotes the
dimension of the square matrix.
square_matrix_algebra
application
This symbol is a binary function, whose first argument should be a
application.
The second argument must be .
entries
application
This symbol is an $(m \times m)$-ary function whose arguments specify
the entries of the square matrix, where $m$ is the dimension.
The square matrix (or block) must be filled row-wise. The number of
arguments MUST match the dimension square of either the matrix algebra
or the surrounding block.
square_matrix
application
This symbol is a binary function whose first argument must be a square
matrix algebra constructor and the second argument is the matrix entries
constructor.
2
2-7
3-1
transpose
application
This symbol is a unary function whose argument should be a matrix.
representation_matrix
application
This symbol is a unary function; its argument is an element b of an
order of a number field. It returns the representation matrix of b
over the field of fractions of its coefficient order.
1
1
2
2
0
2 7
discriminant
application
This symbol is a unary function whose argument is an order of a number
field. It returns the discriminant of the given order. Note that the
argument could also be a number field, and in this case it returns the
discriminant of the maximal order of the given number field.
1
1
2
17
0
trace_matrix
application
This symbol is a unary function whose argument is an order O of a
number field.It returns the matrix containing the traces of the product
of two basis elements of O.
1
1
2
-5
0
determinant
application
This symbol is a unary function whose argument is a square matrix.
It returns the determinant of the given square matrix.
signature
application
This symbol is a unary function whose argument is an order or a number
field. It returns the number of real embeddings the given argument.
1
1
2
9
0
regulator
application
This symbol is a unary function whose argument is an order or a number
field. It returns the regulator of the given argument.
1
1
2
-3
0