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: Integral Basis of a Local Field (ex-9bf4a2)

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

> 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) ]

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