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 LP and a homomorphism φ: L to LP with preimages. The parameter MinPrecision is passed to Completion.
> _<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;
V2.29, 28 November 2025