The complexity of computing all subfields of an algebraic number field

Abstract

For a finite separable field extension K/k, all subfields can be obtained by intersecting so-called principal subfields of K/k. In this work we present a way to quickly compute these intersections. If the number of subfields is high, then this leads to faster run times and an improved complexity.

Functional Decomposition Using Principal Subfields

“…By eorem 2.3, this gives us all complete decompositions of f . Principal subfields and fast field intersection techniques (see [17]) allow us to improve the nonpolynomial part of the complexity.…”

“…By eorem 3.1, the non-polynomial part of the complexity of computing the subfield la ice is then transfered to computing all intersections of the principal subfields. However, according to [17], each subfield of K/k can be uniquely represented by a partition of {1, . .…”

“…. , F r are the irreducible factors of Φ f over K(t), P L represents the factorization of Φ f over L. Algorithms for computing the join of two partitions can be found in [10,17] (see also [11]).…”

“…L , the first P L -product is the minimal polynomial of t over L. As in [17], we give two algorithms for computing the partition of the principal subfield L i : one deterministic and one probabilistic, with be er performance.…”

“…Let f (t) = p(t)/q(t) ∈ K(t), n = max{deg(p), deg(q)} and ∇ f := p(x)q(t) − p(t)q(x) ∈ K[x, t]. Assuming we are given the factorization of ∇ f , using fast arithmetic and fast subfield intersection techniques (see [17]), we can compute the subfield la ice of K(t)/K(f (t)) with an expected number of O(rn 2 ) field operations plusÕ(mr 2 ) CPU operations, where m is the number of subfields of K(t)/K(f (t)) and r ≤ n is the number of irreducible factors of ∇ f (see Corollary 4.16). is approach has the following improvements:…”