Main algorithm

Chapter III: The P -scheme algorithm

3.4 Main algorithm

We present the main body of the P-scheme algorithm in this section. Its input contains a collection of number fields that are isomorphic to subfields of L. In order to avoid duplicate data, we assume that these number fields are mutually non-isomorphic. This is formalized by the following definition:

Definition 3.3((Q, g)-subfield system). Letg(X)be a polynomial inQ[X]with the splitting fieldL(g)overQ. LetF be a collection of number fields such that (1) the fields inF are mutually non-isomorphic, and (2) each field K⁰ ∈ F is isomorphic to a subfield ofL(g). We sayF is a(Q, g)-subfield system.

Given a (Q, g)-subfield system, we define a subgroup system over Gal(g/Q) as follows.

Definition 3.4. Letg(X)be a polynomial inQ[X]with the splitting fieldL(g)over Q. LetF be a(Q, g)-subfield system. DefineP^]to be the poset of subfields ofL(g) that includes all the fields isomorphic to those inF:

P^]:={K⁰ ⊆L(g) :K⁰ ∼=K for someK ∈ F }.

By Galois theory, it corresponds to a posetP of subgroups ofGal(g/Q), given by P :=

H ⊆Gal(g/Q) : (L(g))^H ∈ P^]

which is closed under conjugation in Gal(g/Q), and hence is a subgroup system overGal(g/Q). We sayP andP^]areassociated withF.

The pseudocode of the algorithm is given in Algorithm 2 below. Its input is the prime number p and a (Q,f˜)-subfield system F. We fix P to be the subgroup system overG= Gal( ˜f /Q)associated withF.

The algorithm outputs, for every K ∈ F, the ringO¯_K and an idempotent decom- positionI_K ofO¯_K, together with the auxiliary data of ap-maximal orderO⁰_K and the related maps O_K⁰ ,→ K, O⁰_K → O¯_K. We will see below that the idempotent decompositionsI_K altogether determine aP-collection, which is guaranteed to be a strongly antisymmetricP-scheme when the algorithm terminates.

The first half (Lines 1–7) of the algorithm is the preprocessing stage, where we computeO¯KforK ∈ Fand the ring homomorphisms between them that are induced from the field embeddings. For each K ∈ F, we also initialize the idempotent decompositionI_K ofO¯_K to be the trivial one containing only the unity ofO¯_K. The second half (Lines 8–12) is the “refining” stage. To understand it, we need to associate aP-collectionCwith the idempotent decompositionsI_K. By Lemma 3.12, we knowpsplits completely inF. So it also splits completely in every subfield of L. In particular, for a fieldK inP^]orF, the quotient ringO¯_K is semisimple.

For eachH ∈ P, we define a partitionC_H of the coset spaceH\Gas follows: Let K be the unique field inF isomorphic toL^H. Fix an isomorphismτ_H :K →L^H, which induces a ring isomorphismτ¯_H : ¯O_K →O¯_L^H. DefineI_H := ¯τ_H(I_K), which is an idempotent decomposition of O¯_L^H. By Definition 3.2, it corresponds to a

Algorithm 2ComputePscheme

Input: prime numberp,(Q,f)˜-subfield systemF

Output: for eachK ∈ F: O¯_K, idempotent decompositionI_K ofO¯_K, p-maximal orderO_K⁰ ofK and mapsO_K⁰ ,→K,O⁰_K →O¯_K

1: forK ∈ F do

2: callComputeQuotientRingto computeO¯F, ap-maximal orderO⁰_K ofK and mapsO_K⁰ ,→K,O⁰_K →O¯_K

3: I_K ← {1}, where1denotes the unity ofO¯_K

4: for(K, K⁰)∈ F² do

5: callComputeEmbeddingsto compute all the embeddings fromK toK⁰

6: forembeddingφ :K ,→K⁰ do

7: callComputeRingHomto computeφ¯: ¯O_K →O¯_K⁰ induced fromφ

8: repeat

9: callCompatibilityAndInvarianceTest

10: callRegularityTest

11: callStrongAntisymmetryTest

12: untilI_Kremains the same in the last iteration for allK ∈ F

13: returnO¯_K,I_K,O_K⁰ and the mapsO_K⁰ ,→K,O⁰_K →O¯_K forK ∈ F

partitionP(I_H)ofH\G.7 And we define

C_H :=P(I_H).

Finally, define theP-collectionC by

C :={C_H :H ∈ P}.

We call several subroutines to update I_K in Lines 9–11, whose effects can be understood in terms ofC:

Lemma 3.13. There exists a subroutineCompatibilityAndInvarianceTestthat updates I_K in time polynomial in logp and the size of F so that the partitions C_H ∈ C are refined, and at least one partition C_H is properly refined if C is not compatible or invariant.

7Definition 3.2 is made with respect to a fixed prime idealQ0ofOLlying overp. This ideal is chosen at the beginning of Section 3.3.

Lemma 3.14. There exists a subroutineRegularityTestthat updatesI_K in time polynomial inlogpand the size ofF so that the partitionsC_H ∈ Care refined, and at least one partitionC_H is properly refined ifC is compatible but not regular.

Lemma 3.15. Under GRH, there exists a subroutine StrongAntisymmetryTest that updatesI_K in time polynomial inlogpand the size ofF so that the partitions C_H ∈ C are refined, and at least one partition C_H is properly refined if C is a P-scheme, but not strongly antisymmetric.

We will describe these subroutines and prove the lemmas above in the next three sections. For now we just assume them and prove the main result of this section:

Theorem 3.8(Theorem 3.1 restated). Under GRH, the algorithmComputePscheme runs in time polynomial in the size of the input, and when it terminates, the P- collectionC is a strongly antisymmetricP-scheme.

Proof. We first analyze the running time. As each fieldK ∈ F is encoded by a rational polynomial of degree [K : Q], the total degree N := P

K∈F[K : Q] is bounded by the size ofF. The loops in Lines 1–3 and Lines 4–7 iterate|F | ≤ N and|F²| ≤N²times respectively. For each(K, K⁰)∈ F², there are at most[K :Q] embeddings fromK toK⁰, and hence the inner loop in Lines 6–7 iterates at most [K :Q]times for each fixed(K, K⁰).

For the loop in Lines 8–12, we consider K ∈ F and pick H ∈ P so that L^H is isomorphic to K. By Lemma 3.5, the number of idempotents in I_K equals the number of blocks inC_H, and this number increases every timeI_K is changed by the subroutines. On the other hand, the number of idempotents inI_Kis at most[K :Q]. So the loop in Lines 8–12 iteratesO(N)times. The claim about the running time easily follows.

Finally, note that the algorithm exits the loop in Lines 8–12 after an iteration iff all of the idempotent decompositions I_K remain the same in that iteration, in which caseC is already a strongly antisymmetricP-scheme by Lemma 3.13, Lemma 3.14 and Lemma 3.15.

Remark. The input of the the algorithm containsF whose size may be much greater than that of f andf˜. Therefore, the polynomiality of this algorithm in the size of its input doesnotimply that polynomial factoring over finite fields can be solved in

(deterministic) polynomial time. It does suggest, however, that the total degree of the fields inF overQis the bottleneck of our factoring algorithm.