# IntegralBasisTools(R, UP, F)¶

This package contains functions used in the packages FunctionFieldIntegralBasis and NumberFieldIntegralBasis.

diagonalProduct: Matrix R -> R

diagonalProduct(m) returns the product of the elements on the diagonal of the matrix m

divideIfCan!: (Matrix R, Matrix R, R, Integer) -> R

divideIfCan!(matrix, matrixOut, prime, n) attempts to divide the entries of matrix by prime and store the result in matrixOut. If it is successful, 1 is returned and if not, prime is returned. Here both matrix and matrixOut are n-by-n upper triangular matrices.

idealiser: (Matrix R, Matrix R) -> Matrix R

idealiser(m1, m2) computes the order of an ideal defined by m1 and m2

idealiser: (Matrix R, Matrix R, R) -> Matrix R

idealiser(m1, m2, d) computes the order of an ideal defined by m1 and m2 where d is the known part of the denominator

idealiserMatrix: (Matrix R, Matrix R) -> Matrix R

idealiserMatrix(m1, m2) returns the matrix representing the linear conditions on the Ring associated with an ideal defined by m1 and m2.

leastPower: (NonNegativeInteger, NonNegativeInteger) -> NonNegativeInteger

leastPower(p, n) returns e, where e is the smallest integer such that p ^e >= n

matrixGcd: (Matrix R, R, NonNegativeInteger) -> R

matrixGcd(mat, sing, n) is gcd(sing, g) where g is the gcd of the entries of the n-by-n upper-triangular matrix mat.

moduleSum: (Record(basis: Matrix R, basisDen: R, basisInv: Matrix R), Record(basis: Matrix R, basisDen: R, basisInv: Matrix R)) -> Record(basis: Matrix R, basisDen: R, basisInv: Matrix R)

moduleSum(m1, m2) returns the sum of two modules in the framed algebra F. Each module mi is represented as follows: F is a framed algebra with R-module basis w1, w2, ..., wn and mi is a record [basis, basisDen, basisInv]. If basis is the matrix (aij, i = 1..n, j = 1..n), then a basis v1, ..., vn for mi is given by vi = (1/basisDen) * sum(aij * wj, j = 1..n), i.e. the ith row of ‘basis’ contains the coordinates of the ith basis vector. Similarly, the ith row of the matrix basisInv contains the coordinates of wi with respect to the basis v1, ..., vn: if basisInv is the matrix (bij, i = 1..n, j = 1..n), then wi = sum(bij * vj, j = 1..n).