Efficient enumeration of extensions of local fields with bounded discriminant
Sebastian Pauli · Spectrum Research Repository (Concordia University) · 2001
Let k be a p-adic field. It is well-known that k has only finitely many extensions of a given finite degree. Krasner [1966] gives formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions K/k of a given degree and discriminant. We also present canonical sets of generating polynomials of extensions of degree p^m. Some methods from the proof of the number of extensions of a given degree and discriminant can also be used for the determination of a bound that gives a considerably improved estimate of the complexity of polynomial factorization over local fields. We use this bound in an efficient new algorithm for factoring a polynomial Φ over a local field k. For every irreducible factor φ(x) of Φ(x) our algorithm returns an integral basis for k[x]/φ(x)k[x] over k.