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Local rings can be obtained by completing an order at a
prime ideal (see Chapter NUMBER FIELDS and Completion).
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.
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.
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]);
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