These functions return some simple information which can partially define a local ring or field.
Returns the prime p defining the p-adic ring or field underlying L. This is also the characteristic of the residue class field of the ring L or the ring of integers of the field L.
Returns the inertia degree of the local ring or field L.
Returns the ramification degree of the local ring or field L.
Returns the total degree of the local ring or field L over the underlying p-adic ring or field, i.e. the product of the inertia degree and the ramification degree.
The degree of L over its substructure if L is not a p-adic structure. i.e either the inertia degree if L is an inertia ring or field or the ramification degree otherwise.
Returns the Eisenstein polynomial of L. If L is unramified, the polynomial x - p is returned. The polynomial is over the inertial subring of L.
Returns the minimal polynomial of the inertial element over the p-adic substructure. If the inertial degree of L is one then x - 1 is returned. The polynomial is over the p-adic subring of L.
Returns the minimal polynomial of the generator of L. This will be the Eisenstein polynomial of L unless L is unramified in which case it will return the inertial polynomial of L. This function is not defined if L is p-adic. The polynomial is over the subring of L.
Return the inertia ring of the local ring L.
Return the inertia field of the local field L.
Return the p-adic ring or field which L is an extension of.
Return the finite field isomorphic to the quotient of the local ring L by its maximal ideal pi L (generated by the uniformizing element).
Return the element of valuation 1 the powers of which form the basis of L over its inertial substructure. This will be a root of the Eisenstein polynomial of L. If L is unramified then this is the prime of the ring or field.
Return the inertial element the powers of which form the basis of the inertia ring of L over the p-adic ring. This will be a root of the inertial polynomial of L. If the inertia degree is 1, the 1-element of L is returned.
The precision with which L has been created. If L is a local ring this is the maximum absolute precision its elements can be created with. If L is a local field this is the maximum relative precision its elements can be created with.
> Zp<p> := pAdicRing(5, 20); > I<q, a> := UnramifiedExtension(Zp, 3); > R<x> := PolynomialRing(I); > L<b> := LocalRing(I, x^3 + 5*a*x^2 + 5); > Prime(L); 5 > InertiaDegree(L); 3 > TotalDegree(L); 9 > EisensteinPolynomial(L); x^3 + 5*a*x^2 + 5 > P<y> := PolynomialRing(Zp); > InertialPolynomial(L); y^3 + 11224679742905*y^2 - 36114878820502*y + 15613890344818 > DefiningPolynomial(L); x^3 + 5*a*x^2 + 5 > DefiningPolynomial(I); y^3 + 11224679742905*y^2 - 36114878820502*y + 15613890344818 > InertiaRing(L); Inertia Ring of degree 3 over Zp > PrimeRing(L); 5-adic Ring mod p^20 > PrimeRing(I); 5-adic Ring mod p^20 > ResidueClassField(L); Finite field of size 5^3 Mapping from: RngLoc: L to GF(5^3) > ResidueClassField(I); Finite field of size 5^3 Mapping from: RngLoc: I to GF(5^3) > UniformizingElement(L); b > L.2; a > Precision(L); 60 > Precision(I); 20 > R<y> := PolynomialRing(Rationals()); > R<b, a> := LocalRing(2, y^2 + y + 1); > InertialPolynomial(R); $.1^2 + $.1 + 1 > Precision(R); InfinityThe printing of the polynomials can be possibly made both messier and more informative by turning the SeriesPrinting attribute on for the p-adic substructure.
> Zp`SeriesPrinting := true;
> InertialPolynomial(L);
y^3 + (p + p^2 - 2*p^3 - p^4 - 2*p^5 - p^6 + p^7 + p^9 + 2*p^10 + p^11 + p^12 -
p^14 - 2*p^15 - p^16 - 2*p^18 + p^19)*y^2 + (-2 + p^3 + 2*p^4 + 2*p^5 + p^7
+ 2*p^9 + p^10 + 2*p^11 - 2*p^12 - 2*p^14 + 2*p^15 - 2*p^16 - 2*p^17 + p^18
- 2*p^19)*y - 2 - p - 2*p^2 - p^3 + 2*p^4 + 2*p^6 + p^7 - p^8 + 2*p^9 +
2*p^10 + 2*p^11 - p^12 + p^13 - 2*p^14 + 2*p^15 + 2*p^16 - p^18 + p^19
> DefiningPolynomial(L);
x^3 + q*a*x^2 + q