# Completion

## `Completion(P): AlgEtQIdl -> FldPad, Map`

```magma
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)`

```magma
> _<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;

```