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 [Pauli, 2001] developed explicit methods to enumerate those extensions.

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

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

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

Given a \(p\)-adic ring or field \(R\) and some positive integer \(n\), compute all the extensions of \(R\) of degree \(n\). At least one extension is given in every isomorphism class. The optional parameters can be used to impose restrictions on the fields returned. Note that \(j\) and \(vD\) must be unspecified (as \(-1\)) when \(F\neq 1\) or when \(E\) and \(F\) are both not given.

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.

vD

specifies the valuation of the discriminant, \(-1\) implies no restriction.

j

specifies the valuation of the discriminant via the formula \(vD := n+j-1\).

Galois

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

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

E     : RngIntElt                    Default: 0
F     : RngIntElt                    Default: 0
Galois: BoolElt                      Default: false
vD    : RngIntElt                    Default: -1
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. Note that \(j\) and \(vD\) must be unspecified (as \(-1\)) when \(F\neq 1\) or when \(E\) and \(F\) are both not given.

Note that the count will not be the same as AllExtensions, as the latter need only be up to isomorphism.

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 exist totally ramified extensions of \(R\) of degree \(n\) with discriminant valuation \(n+j-1\).

Example: All Extensions (ex-ea2536)

We follow Examples 9.1 and 9.2 from [Pauli and Roblot, 2001].

There are 54 (totally ramified) extensions of degree 9 and discriminant \(3^{9+4-1}\) over \({{\mathbb{Q}}}_3\). There are six generating polynomials, each defining nine isomorphism classes. The possible (nontrivial) subfields of these have degree 3 and \(j_0=1\), of which there are two defining polynomials each with three isomorphism classes. Each of these degree 3 fields then admits two extensions with \(j_1=1\), which give six isomorphism classes. This gives a total of 27 degree 9 extensions that have a subfield of degree 3.

> R := pAdicRing(3,20);
> _<x> := PolynomialRing(R); // for printing
> NumberOfExtensions(R,9 : F:=1,j:=4);
54
> A9 := AllExtensions(R,9 : F:=1,j:=4);
> [DefiningPolynomial(a) : a in A9];
[ x^9 + 3*x^4 + 3, x^9 + 6*x^4 + 3,
  x^9 + 3*x^4 + 3*x^3 + 3, x^9 + 6*x^4 + 3*x^3 + 3,
  x^9 + 3*x^4 + 6*x^3 + 3, x^9 + 6*x^4 + 6*x^3 + 3 ]
> A3 := AllExtensions(R,3 : F:=1,j:=1);
> NumberOfExtensions(A3[1],3 : F:=1,j:=1);
6
> [DefiningPolynomial(a) : a in A3];
[ x^3 + 3*x + 3, x^3 + 6*x + 3 ]
> _<pi> := A3[1];
> _<y> := PolynomialRing(A3[1]);
> B3 := AllExtensions(A3[1],3 : F:=1,j:=1);
> [DefiningPolynomial(f) : f in B3];
[ y^3 + pi*y + pi, y^3 + 2*pi*y + pi ]

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The other example concerns degree 10 extensions of \({{\mathbb{Q}}}_5\). Here there are 1818 total extensions, of which 1 is unramified and 2 have ramification degree 2, while 605 have ramification degree 5 and 1210 are totally ramified. With ramification degree 5, there are 145 defining polynomials over the unramified quadratic field, split into five \(j\)-groupings. As noted in [Pauli and Roblot, 2001], there is a further splitting in the \(j=4\) grouping. Similarly, there are 145 defining polynomials over either of the two tamely ramified extensions of degree 2 over \({{\mathbb{Q}}}_5\). The resulting fields are in fact isomorphic in pairs, but the AllExtensions function still lists both fields in each pair. So it returns 438 fields (1+2+145+290) rather than the stated 293 isomorphism classes (1+2+145+145).

> R := pAdicRing(5,20);
> NumberOfExtensions(R,10);
1818
> [NumberOfExtensions(R,10 : E:=e) : e in Divisors(10)];
[ 1, 2, 605, 1210 ]
> U := UnramifiedExtension(R,2);
> [#AllExtensions(U,5 : E:=5,j:=j0) : j0 in [1..5]];
[ 24, 24, 24, 48, 25 ]
> // compare the above/below to (#K)/N in Pauli-Roblot
> [#AllExtensions(R,10 : E:=10,j:=j0): j0 in [1..10]];
[ 8, 8, 8, 16, 0, 40, 40, 80, 40, 50 ] // twice P-R

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