Efficient enumeration of extensions of local fields with bounded discriminant

• Main Author: Pauli, Sebastian
• Format: Others
• Published: 2001
• Online Access: http://spectrum.library.concordia.ca/1533/1/NQ63983.pdf; Pauli, Sebastian <http://spectrum.library.concordia.ca/view/creators/Pauli=3ASebastian=3A=3A.html> (2001) Efficient enumeration of extensions of local fields with bounded discriminant. PhD thesis, Concordia University.

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 Z over a local field k . For every irreducible factor [varphi]( x ) of Z ( x ) our algorithm return an integral basis for k [ x ]/[varphi]( x ) k [ x ] over k .