# Maximal Order

## `IntegralBasis(L): RngLocA -> SeqEnum`

Return a basis for the maximal order of the local field $L$.

## `IsIntegral(a): RngLocAElt -> BoolElt, SeqEnum`

Return whether the local field element $a$ lies in the maximal order of its parent $L$ and a sequence giving the coordinates of $a$ with respect to the integral basis of $L$ if so.

## `Example: Max Order (ex-9bf4a2)`

We construct a local field and compute an integral basis for it.

```magma
> P<x> := PolynomialRing(Integers());
> L := LocalField(pAdicField(7, 50), x^6 - 49*x^2 + 686);
> IntegralBasis(L);
[ 1 + O(7^50), (7^-1 + O(7^49))*$.1^2 + O(7^49)*$.1 + O(7^49), (7^-2 +
    O(7^48))*$.1^4 + O(7^48)*$.1^3 + O(7^48)*$.1^2 + O(7^50)*$.1 + O(7^50), $.1
    + O(7^50), (7^-1 + O(7^49))*$.1^3 + O(7^49)*$.1^2 + O(7^49)*$.1 + O(7^99),
    (7^-2 + O(7^48))*$.1^5 + O(7^48)*$.1^4 + O(7^48)*$.1^3 + O(7^50)*$.1^2 +
    O(7^50)*$.1 + O(7^50) ]

```