padicfields

Function: padicfields
Section: polynomials
C-Name: padicfields0
Prototype: GGD0,L,
Help: padicfields(p, N, {flag=0}): returns polynomials generating all
 the extensions of degree N of the field of p-adic rational numbers; N is
 allowed to be a 2-component vector [n,d], in which case, returns the
 extensions of degree n and discriminant p^d. flag is optional,
 and can be 0: default, 1: return also the ramification index, the residual
 degree, the valuation of the discriminant and the number of conjugate fields,
 or 2: return only the number of extensions in a fixed algebraic closure.

Doc: returns a vector of polynomials generating all the extensions of degree
 $N$ of the field $\Q_p$ of $p$-adic rational numbers; $N$ is
 allowed to be a 2-component vector $[n,d]$, in which case we return the
 extensions of degree $n$ and discriminant $p^d$.

 The list is minimal in the sense that two different polynomials generate
 nonisomorphic extensions; in particular, the number of polynomials is the
 number of classes of nonisomorphic extensions. If $P$ is a polynomial in this
 list, $\alpha$ is any root of $P$ and $K = \Q_p(\alpha)$, then $\alpha$
 is the sum of a uniformizer and a (lift of a) generator of the residue field
 of $K$; in particular, the powers of $\alpha$ generate the ring of $p$-adic
 integers of $K$.

 If $\fl = 1$, replace each polynomial $P$ by a vector $[P, e, f, d, c]$
 where $e$ is the ramification index, $f$ the residual degree, $d$ the
 valuation of the discriminant, and $c$ the number of conjugate fields.
 If $\fl = 2$, only return the \emph{number} of extensions in a fixed
 algebraic closure (Krasner's formula), which is much faster.

Variant: Also available is
 \fun{GEN}{padicfields}{GEN p, long n, long d, long flag}, which computes
 extensions of $\Q_p$ of degree $n$ and discriminant $p^d$.