Extensions of Series Rings

Extensions of series rings are either unramified or totally ramified. Only series rings defined over finite fields can be extended. We recommend constructing polynomials from sequences rather than by addition of terms, especially over fields, to avoid some precision loss. For example, x⁴ + t will not have full precision in the constant coefficient as there will be a O term in the constant coefficient of x⁴ which will reduce its precision as a field element. Extensions require full precision polynomials and some polynomials such as x⁴ + t may not have enough precision to be used to construct an extension whereas the equivalent Polynomial([t, 0, 0, 0, 1]) will.

Constructions of Extensions

UnramifiedExtension(R, f) : RngSerPow[FldFin], RngUPolElt -> RngSerExt

UnramifiedExtension(R, f) : RngSerLaur[FldFin], RngUPolElt -> RngSerExt

UnramifiedExtension(R, f) : RngSerExt, RngUPolElt -> RngSerExt

Construct the unramified extension of R, a series ring or an extension thereof, defined by the inertial polynomial f, that is, adjoin a root of f to R.

TotallyRamifiedExtension(R, f) : RngSerPow[FldFin], RngUPolElt -> RngSerExt

TotallyRamifiedExtension(R, f) : RngSerLaur[FldFin], RngUPolElt -> RngSerExt

TotallyRamifiedExtension(R, f) : RngSerExt, RngUPolElt -> RngSerExt

    MaxPrecision: RngIntElt             Default: 

Construct a totally ramified extension of R, a series ring or an extension thereof, defined by the eisenstein polynomial f, that is, adjoin a root of f to R.

The parameter MaxPrecision defaults to the precision of R. It can be set to the maximum precision the coefficients of f are known to, which must not be less than the precision of R. This allows the precision of the result to be increased to the degree of f multiplied by this maximum precision.

The polynomial f may be given over a series ring or an extension of a series ring having a higher precision than R. This allows the precision of the result to be increased up to the precision the polynomial is known to (or MaxPrecision if set) without losing any of the polynomial known past the precision of R.

The precision of a ramified extension cannot be increased unless the defining polynomial is known to more precision than the coefficient ring, indicated either by providing the polynomial to greater precision or by setting MaxPrecision. This should be taken into account when constructing ramified extensions, especially if polynomials are to be factored over the extension.

ChangePrecision(E, r) : RngSerExt, RngIntElt -> RngSerExt

ChangePrecision(~E, r) : RngSerExt, RngIntElt ->

The extension of a series ring E with precision r. It is not possible to increase the precision of a ramified extension unless the parameter MaxPrecision was set on construction of the extension and r is less than or equal to this value multiplied by the ramification degree or the polynomial used in creating the extension was given to more precision than the coefficient ring of E.

FieldOfFractions(E) : RngSerExt -> RngSerExt

The field of fractions of the extension of a series ring E.

> P<t> := PowerSeriesRing(GF(101), 50);
> PP<tt> := PowerSeriesRing(GF(101));
> R<x> := PolynomialRing(PP);
> U := UnramifiedExtension(P, x^2 + 2);
> T := TotallyRamifiedExtension(U, x^2 + tt*x + tt); T;
Extension of Extension of Power series ring in t over GF(101) with fixed 
absolute precision 50 by x^2 + 2 + O(t^50) by x^2 + (t + O(t^50))*x + t + 
    O(t^50)
> Precision($1);
100
> ChangePrecision($2, 200);
Extension of Extension of Power series ring in $.1 over GF(101) with fixed 
absolute precision 100 by x^2 + 2 + O($.1^100) by x^2 + ($.1 + O($.1^100))*x + 
    $.1 + O($.1^100)
> ChangePrecision($1, 1000);
Extension of Extension of Power series ring in $.1 over GF(101) with fixed 
absolute precision 500 by x^2 + 2 + O($.1^500) by x^2 + ($.1 + O($.1^500))*x + 
    $.1 + O($.1^500)
> ChangePrecision($1, 20);      
Extension of Extension of Power series ring in $.1 over GF(101) with fixed 
absolute precision 10 by x^2 + 2 + O($.1^10) by x^2 + ($.1 + O($.1^10))*x + $.1 
    + O($.1^10)

> FieldOfFractions(U);
Extension of Laurent series field in $.1 over GF(101) with fixed relative 
precision 50 by (1 + O($.1^50))*x^2 + O($.1^50)*x + 2 + O($.1^50)
> FieldOfFractions(T);
Extension of Extension of Laurent series field in $.1 over GF(101) with fixed 
relative precision 50 by (1 + O($.1^50))*x^2 + O($.1^50)*x + 2 + O($.1^50) by (1
    + O($.1^50))*x^2 + ($.1 + O($.1^50))*x + $.1 + O($.1^50)

> L<t> := LaurentSeriesRing(GF(53), 30); 
> P<x> := PolynomialRing(L);
> U := UnramifiedExtension(L, x^3 + 3*x^2 + x + 4);
> P<y> := PolynomialRing(U);
> T := TotallyRamifiedExtension(U, Polynomial([t, 0, 0, 0, 1]));
> Precision(U);
30
> Precision(T);
120
> CoefficientRing(U);
Laurent series field in t over GF(53) with fixed relative precision 30
> CoefficientRing(T);
Extension of Laurent series field in t over GF(53) with fixed relative precision
30 by (1 + O(t^30))*x^3 + (3 + O(t^30))*x^2 + (1 + O(t^30))*x + 4 + O(t^30)
> DefiningPolynomial(U);
(1 + O(t^30))*x^3 + (3 + O(t^30))*x^2 + (1 + O(t^30))*x + 4 + O(t^30)
> DefiningPolynomial(T);
(1 + O(t^30))*y^4 + O(t^30)*y^3 + O(t^30)*y^2 + O(t^30)*y + t + O(t^30)
> InertiaDegree(U);
3
> InertiaDegree(T);
1
> RamificationDegree(U);
1
> RamificationDegree(T);
4
> ResidueClassField(U);
Finite field of size 53^3
Mapping from: RngSerExt: U to GF(53^3)
> ResidueClassField(T);
Finite field of size 53^3
Mapping from: RngSerExt: T to GF(53^3)
> UniformizingElement(U);
t + O(t^31)
> UniformizingElement(T);
(1 + O(t^30))*$.1 + O(t^30)
> Integers(T);
Extension of Extension of Power series ring in $.1 over GF(53) with fixed 
absolute precision 30 by x^3 + (3 + O($.1^30))*x^2 + x + 4 + O($.1^30) by x^4 + 
    $.1 + O($.1^30)
> U.1;
(1 + O(t^30))*$.1 + O(t^30)
> T.1;
(1 + O(t^30))*$.1 + O(t^30)

> P<t> := PowerSeriesRing(GF(101), 50);
> R<x> := PolynomialRing(P);
> U<u> := UnramifiedExtension(P, x^2 + 2*x + 3);
> UF := FieldOfFractions(U);
> R<y> := PolynomialRing(U);
> T<tt> := TotallyRamifiedExtension(U, y^2 + t*y + t);
> TF<tf> := FieldOfFractions(T);
> UF<uf> := FieldOfFractions(U);                      
> u + t;
u + t + O(t^50)
> uf * t;
($.1 + O($.1^50))*uf + O($.1^51)
> tf eq tt;
false
> tf - tt; 
O($.1^50)*tf + O($.1^50)
> IsZero($1);
false
> Valuation($2);
100
> Valuation(tt); 
1
> Valuation(U!t);
1
> Valuation(T!t);
2
> RelativePrecision(u);
50
> AbsolutePrecision(u);
50
> AbsolutePrecision(uf);
50
> RelativePrecision(uf);
50
> RelativePrecision(u - uf);
0
> AbsolutePrecision(u - uf); 
50
> u^7;
(13 + O(t^50))*u + 71 + O(t^50)
> Coefficients($1);
[
    71 + O(t^50),
    13 + O(t^50)
]
> tt^8;
(4*t^4 + 91*t^5 + 6*t^6 + 100*t^7 + O(t^50))*tt + t^4 + 95*t^5 + 5*t^6 + 100*t^7
    + O(t^50)
> Coefficients($1);
[ t^4 + 95*t^5 + 5*t^6 + 100*t^7 + O(t^50), 4*t^4 + 91*t^5 + 6*t^6 + 100*t^7 + 
    O(t^50) ]

> P<t> := PowerSeriesRing(GF(101), 50);
> R<x> := PolynomialRing(P);
> U<u> := UnramifiedExtension(P, x^2 + 2*x + 3);
> U;
Extension of Power series ring in t over GF(101) with fixed absolute precision 
50 by x^2 + (2 + O(t^50))*x + 3 + O(t^50)
> OptimizedRepresentation(U);                                                  
Power series ring in $.1 over GF(101^2) with fixed absolute precision 50
Mapping from: RngSerExt: U to Power series ring in s over GF(101^2) with fixed 
absolute precision 50 given by a rule