# PAdicWildFunctionFieldIntegralBasis(K, R, UP, F)ΒΆ

- K: FiniteFieldCategory
- R: UnivariatePolynomialCategory K
- UP: UnivariatePolynomialCategory R
- F: MonogenicAlgebra(R, UP)

In this package `K`

is a finite field, `R`

is a ring of univariate polynomials over `K`

, and `F`

is a monogenic algebra over `R`

. We require that `F`

is monogenic, i.e. that `F = K[x, y]/(f(x, y))`

, because the integral basis algorithm used will factor the polynomial `f(x, y)`

. The package provides a function to compute the integral closure of `R`

in the quotient field of `F`

as well as a function to compute a “local integral basis” at a specific prime.

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

returns a record`[basis, basisDen, basisInv] ``\ containing information regarding the integral closure of \ ``R`

in the quotient field of the framed algebra`F`

.`F`

is a framed algebra with`R`

-module basis`w1, w2, ..., wn`

. If ‘basis’ is the matrix`(aij, i = 1..n, j = 1..n)`

, then the`i`

th element of the integral basis is`vi = (1/basisDen) * sum(aij * wj, j = 1..n)`

, i.e. the`i`

th row of ‘basis’ contains the coordinates of the`i`

th basis vector. Similarly, the`i`

th 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)`

.

- localIntegralBasis: R -> Record(basis: Matrix R, basisDen: R, basisInv: Matrix R)
`integralBasis(p)`

returns a record`[basis, basisDen, basisInv] ``\ containing information regarding the local integral closure of \ ``R`

at the prime`p`

in the quotient field of the framed algebra`F`

.`F`

is a framed algebra with`R`

-module basis`w1, w2, ..., wn`

. If ‘basis’ is the matrix`(aij, i = 1..n, j = 1..n)`

, then the`i`

th element of the local integral basis is`vi = (1/basisDen) * sum(aij * wj, j = 1..n)`

, i.e. the`i`

th row of ‘basis’ contains the coordinates of the`i`

th basis vector. Similarly, the`i`

th 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)`

.

- reducedDiscriminant: UP -> R
`reducedDiscriminant(up)`

undocumented