MAGMA Computational Algebra System

Extensions

It is a well known classical theorem that p-adic fields admit only finitely many different extensions of bounded degree (in contrast to number fields which have an infinite number of extensions of any degree). In his thesis, Pauli [Pau01a] developed explicit methods to enumerate those extensions.

AllExtensions(R, n) : RngPad, RngIntElt -> [RngPad]

AllExtensions(R, n) : FldPad, RngIntElt -> [RngPad]

    E : RngIntElt : 0
   var F : RngIntElt : 0
   var Galois : BoolElt :false
   var D : RngIntElt : 0
   var j: RngIntElt Default: -1

Given a p-adic ring or field R and some positive integer n, compute defining equations for all extensions of R of degree n. The optional parameters can be used to limit the extensions in various ways:

E specifies the ramification index. 0 implies no restriction.

F specifies the inertia degree, 0 implies no restriction.

D specified the valuation of the discriminant, 0 implies no restriction.

j specifies the valuation of the discriminant via the formula D := d + j - 1 where d is the degree of R.

Galois when set to true, limits the extensions to only list normal extensions.

NumberOfExtensions(R, n) : RngPad, RngIntElt -> RngIntElt

    E : RngIntElt : 0
   var F : RngIntElt : 0
   var Galois : BoolElt :false
   var D : RngIntElt : 0
   var j: RngIntElt Default: -1

Given a p-adic ring or field R and some positive integer n, compute the number of extensions of R of degree n. Similarly to the above function, the optional parameters can be used to impose restrictions on the fields returned.

OreConditions(R, n, j) : RngPad, RngIntElt, RngIntElt -> BoolElt

OreConditions(R, n, j) : FldPad, RngIntElt, RngIntElt -> BoolElt

Given a p-adic ring or field R and positive integers n and j, test if there exists extensions of R of degree n with valuation of the discriminant being n + j - 1.