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