At the moment we only allow a two-step definition of local fields.

Subsections

• Precision

Given a prime p and integers f and m, create the local field obtained by a residue class field extension of degree f of Q_p, to precision m. Angle brackets may be used to assign two `indeterminate names'; the first one will be a multiplicative representative of the residue class field extension in the local field, the second one will refer to the prime p from Q_p in the local field.

UnramifiedExtension(Qp, f, m) : FldLoc, RngIntElt -> FldLoc

Given a `free' p-adic field Q_p and positive integers f, m, create the local field obtained by a residue class field extension of degree f, to precision m. Angle brackets may be used to assign two `indeterminate names'; the first one will be a multiplicative representative of the residue class field extension in the local field, the second one will refer to the prime p from Q_p in the local field.

UnramifiedExtension(Qp, f) : FldLoc, RngIntElt -> FldLoc

Given a p-adic field Q_p of fixed precision m, and a positive integer f, create the local field obtained by a residue class field extension of degree f, to precision m. Angle brackets may be used to assign two `indeterminate names'; the first one will be a multiplicative representative of the residue class field extension in the local field, the second one will refer to the prime p from Q_p in the local field.

Given an unramified extension F of Q_p, and an Eisenstein polynomial f over F, create the totally ramfied extension F[x]/f(x). Note that an Eisenstein polynomial of degree e over F is a monic irreducible polynomial x^e + c_(e - 1) x^(e - 1) + ... + c_1x + c_0 such that c_0 in F has valuation 1 (that is, is exactly divisible by p) and c_i has valuation at least 1, that is, all c_i are divisible by p (for i=1, ..., e - 1).

Note that it will be necessary at present to create a trivial extension of the p-adics first if residue class degree f=1 is desired.

Given a prime number p and a univariate monic polynomial P over the integers or the rationals that is irreducible mdulo p, create the local field Q_p[x]/P(x).

[Future release] LocalField(Qp, P) : FldLoc, URngPolElt -> FldLoc

[Future release] ext<Qp | P > : FldLoc, URngPolElt -> FldLoc

Given a p-adic field Q_p and a univariate monic irreducble polynomial P over Q_p, create the local field Q_p[x]/P(x).

Upon creation of a ring of series it is possible to set the default precision by using Precision as an optional argument. Once the ring has been created the default precision for elements can be chamged by using AssertAttribute.

AssertAttribute(A, "Precision", n) : AlgPow, MonStgElt, RngIntElt ->

Given a ring of (Laurent or power) series A, this procedure changes the default precision on elements created to n.