Completions

Local rings can be obtained by completing an order at a prime ideal (see Chapter Number Fields and Completion).

Completion(O, P): RngOrd, RngOrdIdl -> RngPad, Map

Completion(K, P): FldNum, RngOrdIdl -> FldPad, Map

Precision: RngIntElt                    Default: 20

The completion (as an unbounded precision local ring or field with default precision given by Precision) of the order or number field at the prime ideal \(P\), and the embedding of the order or number field into the resulting local ring.

LocalRing(P, k): RngOrdIdl, RngIntElt -> RngPad, Map: The completion (as a local ring) of the order of the prime ideal \(P\) at \(P\) with default precision \(k\) and the embedding of the order into the resulting local ring.

Example: Completion of a Number Field (ex-2bb9ae)

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Here we demonstrate the use of Completion.

> K := NumberField(x^6 - 5*x^5 + 31*x^4 - 85*x^3 + 207*x^2 - 155*x + 123);
> lp := Decomposition(K, 7);
> C, mC := Completion(K, lp[2][1]);
> C;
Totally ramified extension defined by a map over Unramified extension defined by
a map over 7-adic field
> mC;
Mapping from: FldNum: K to FldPad: C given by a rule
> mC(K.1);
(46564489*$.1 - 47959419)*C.1 - 116434149*$.1 - 61099304 + O(C.1^20)
> delta := (K.1 @ mC @@ mC) - K.1;
> delta;
8337821493402521350488*K.1^5 - 69073506960056896464432*K.1^4 +
    189847416443444330877726*K.1^3 - 453361530291976951337876*K.1^2 +
    336979647814116799276099*K.1 - 267520869714197002579071
> // Check the accuracy of the mappings using the valuation of the difference
> Valuation(delta, lp[2][1]);
18
> C`DefaultPrecision := 30;
> mC(K.1);
(1090965976127*$.1 - 1208477074641)*C.1 - 589359803563*$.1 + 288063654676 +
O(C.1^30)
> delta := (K.1 @ mC @@ mC) - K.1;
> delta;
-61980024244160371672868773433490783*K.1^5 +
    1189796803064803092593291088768968754*K.1^4 -
    3202353946933190588864309180653868957*K.1^3 +
    7537386928046164580731145031872017049*K.1^2 -
    5511297002936682579964210586013308810*K.1 +
    4438099444806431313582533435941098722
> Valuation(delta, lp[2][1]);
28
> C`DefaultPrecision := 10;
> mC(K.1);
(-7708*$.1 + 7759)*C.1 + 4747*$.1 - 5859 + O(C.1^10)
> delta := (K.1 @ mC @@ mC) - K.1;
> delta;
1908210240*K.1^5 - 7326424608*K.1^4 + 16701662320*K.1^3 - 35965440540*K.1^2 +
    41324075079*K.1 - 30476856505
> Valuation(delta, lp[2][1]);
8

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Completions#

Completions