On the computation of all extensions of a 𝑝-adic field of a given degree

Abstract

Abstract. Let k be a p-adic field. It is well-known that k has only finitely many extensions of a given finite degree. Krasner has given 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.

Generalized Explicit Descent and Its Application to Curves of Genus 3

“…On the reduction of X at v, the 28 bitangents are distinct, so the reduction of (x : y : z) ∈ X (k v ) lies on at most one bitangent, so v(u Q x + v Q y + w Q z) > 0 for at most one Q. On the other hand, (27) and our hypothesis on (u θ , v θ , w θ ) imply…”

Octic 2-adic fields

“…Let K e,f,c be the set of isomorphism classes of octic 2-adic fields with residual degree f , ramification degree e, and discriminantal ideal (2 c ). General p-adic mass formulas, [12,20,15], give the total mass of K e,f,c . The set K 1,8,0 has one element, the unramified octic extension of Q 2 ; so its total mass is 1/8.…”

“…Once one knows the mass of K e,f,c , there is a procedure for obtaining a defining polynomial f i (x) for each element of K e,f,c [15]. Carrying out this procedure, we find the desired f i (x).…”

Constructing class fields over local fields

“…There are formulas [Kra66,PR01] for the number of extensions of a p-adic field of a given degree and discriminant given by: Theorem 4.1 (Krasner). Let K be a finite extension of Q p , and let j = aN + b, where 0 b < N , be an integer satisfying Ore's conditions: Let j = ap + b satisfy Ore's conditions for ramified extensions of degree p then…”