Completion

Completion(P): AlgEtQIdl -> FldPad, Map

MinPrecision: RngIntElt                    Default: 20

Given a prime ideal \(P\) of the maximal order of an étale algebra \(L\), returns the \(p\)-adic field corresponding to the completion \(L_P\) and a homomorphism \(\phi: L\to L_P\) with preimages. The parameter MinPrecision is passed to Completion.

Example: Uniformizers Completion (ex-ecd376)

> _<x> := PolynomialRing(Integers());
> f := (x^8+16)*(x^8+81);
> A := EtaleAlgebra(f);
> O := MaximalOrder(A);
> // Consider a bunch of prime of O and their uniformizers in O.
> pp := PrimesAbove(2*3*5*7*O);
> unifs := Uniformizers(pp);
> // We now verify that each element is a uniformizer at the correct prime
> // and a unit everywhere else
> assert Matrix([[ Valuation(mP(t)) : t in unifs ] where AP, mP := Completion(P)
>                                                                : P in pp]) eq 1;

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