Reduction to primitive group actions

Chapter IV: Constructing number fields

4.3 Reduction to primitive group actions

of the loop in Lines 5–12, every field in F is isomorphic to some field inP_k over K₀ and vice versa. This follows from a simple induction onk.

Denote by dthe maximum degree of the fields inF overK₀. Then dand|F |are bounded by c(P). By induction and Lemma 4.8, each field in F is encoded by the minimal polynomial of a primitive element k₁α₁ +· · ·+k_sα_s overK₀ where s ≤ m ≤ deg(g), all α_i are roots of g, and1 ≤ k_i ≤ d+ 1. The claim about the running time then follows from Lemma 4.4 and Lemma 4.8.

By Lemma 4.2, the complexity c(P)of the subgroup system P in Lemma 4.10 is bounded by(deg(g))^m0, where m⁰ = min{deg(g), m}. Lemma 3.21 then follows by choosingK₀ =Q.

Lemma 4.11. Let S be a finite set where |S| > 1, and let x ∈ S. A transitive permutation groupGonSis primitive iffG_xis maximal inG.

See, e.g., [Wie64] for the proof of Lemma 4.11. We also need the following result, proved in [LM85].

Theorem 4.1([LM85]). There exists a polynomial-time algorithmTowerthat given a number fieldK₀and a polynomialg(X)∈K₀[X]irreducible overK₀,5computes a tower of relative number fields overK0

K₀ ⊆K₁ ⊆ · · · ⊆K_k−1 ⊆K_k

together with the inclusions Ki−1 ,→ K_i and the polynomials g_i(X) ∈ Ki−1[X]

irreducible over Ki−1 for i ∈ [k], such that K_k ∼=K0 K₀[X]/(g(X)), and the following conditions are satisfied fori∈[k]:

1. K_iis isomorphic toKi−1[X]/(g_i(X))overKi−1, and

2. the Galois groupG_i := Gal(L_i/K_i−1)acts primitively on the set of roots of g_i inL_i, whereL_i is the Galois closure ofK_i/K_i−1.

Fori ∈ [k], letHi := Gal(Li/Ki) ⊆ Gi. See Figure 4.1 for an illustration. Note that the first condition above is equivalent to K_i = K_i−1(α) for some root α_i of g_i inL_i. So H_i is the stabilizer ofα_i. Then the second condition is equivalent to maximality ofH_i inG_i.

The following theorem is the main result of this section.

Theorem 4.2. Suppose there exists an algorithm PrimitiveAction that, given a number field K₀ and a polynomial g(X) ∈ K₀[X] irreducible over K₀ with Gal(g/K₀)acting primitively on the set of roots ofg inL, whereLis the splitting field ofg overK₀, computes a(K₀, g)-subfield system in timeT(K₀, g). Then there exists an algorithmGeneralActionthat givenK₀ andg as above, butwithoutthe assumption thatGal(g/K₀)acts primitively onS, computes

• a(K₀, g)-subfield systemF, and,

5The paper [LM85] presented their algorithm only forK₀=Q, but it easily extends to a general base fieldK0.

K₀ K₁

K₂

· · · Kk−1

K_k

L₁ L₂

L_k

G₁ H₁ G₂ H₂

G_k H_k

Figure 4.1: The tower of fields and Galois groups in Theorem 4.1

• a tower of relative number fields K₀ ⊆ K₁ ⊆ · · · ⊆ Kk−1 ⊆ K_k over K₀ and g_i(X) ∈ Ki−1[X] for i ∈ [k] satisfying the conditions in Theorem 4.1, such that Kk ∼=K0 K0[X]/(g(X)) and the sizes of the polynomials gi are polynomial in the size of the input

in time polynomial inPk

i=1T(Ki−1, g_i)and the size of the input. Moreover, if for eachi ∈ [k], the(Ki−1, g_i)-subfield systemF_i computed by PrimitiveActionon the input(Ki−1, g_i)satisfies

1. Ki−1[X]/(g_i(X))∈ F_i,

2. All strongly antisymmetricP-schemes are discrete (resp. inhomogeneous) on H, wherePis the subgroup system overGal(gi/Ki−1)associated withFiand His a subgroup inP whose fixed field is isomorphic toKioverKi−1. ThenF satisfies

1. K₀[X]/(g(X))∈ F,

2. All strongly antisymmetricP-schemes are discrete (resp. inhomogeneous) on H, where P is the subgroup system over Gal(g/K₀)associated withF and His a subgroup inP satisfyingL^H ∼=K0 K₀[X]/(g(X)).

See Algorithm 9 for the pseudocode of the algorithmGeneralAction. It proceeds as follows: maintain F, which initially only contains K0[X]/(g(X)). Then we

call the algorithm Tower to compute a tower K₀ ⊆ K₁ ⊆ · · · ⊆ K_k−1 ⊆ K_k and g_i(X) ∈ K_i−1[X] for i ∈ [k] as in Theorem 4.1. Next, run the hypothetical algorithmPrimitiveActionin Theorem 4.2 on(Ki−1, g_i)for eachi∈[k]to obtain a(Ki−1, g_i)-subfield system F_i. Fori∈ [k], add the fields inF_i to F, but encode them as relative number fields overK₀(using Lemma 4.6). In addition, avoid adding fields toF that are isomorphic to some existent fieldK ∈ F overK₀, so that all the fields inFare mutually non-isomorphic overK₀. After allF_iare processed, output F.

Algorithm 9GeneralAction

Input: number fieldK₀ andg(X)∈K₀[X]irreducible overK₀ Output: (K₀, g)-subfield systemF

1: F ← {K₀[X]/(g(X))}

2: run Tower on(K₀, g) to obtain a tower K₀ ⊆ K₁ ⊆ · · · ⊆ Kk−1 ⊆ K_k and g_i(X)∈Ki−1[X]irreducible overKi−1fori∈[k]

3: fori←1tokdo

4: runPrimitiveActionon(Ki−1, g_i)to obtainF_i

5: forK ∈ F_i do

6: compute a relative number fieldK⁰ overK0 such thatK⁰ ∼=K0 K

7: if K⁰ is non-isomorphic to all fields inF overK₀ then

8: F ← F ∪ {K⁰}

9: returnF

The proof of Theorem 4.2 is based on the following lemma.

Lemma 4.12. Let k ∈ N⁺ and G_k ⊆ Gk−1 ⊆ · · · ⊆ G₁ ⊆ G₀ be a chain of finite groups. For i ∈ [k], let N_i be a subgroup of G_i that is normal in Gi−1, π_i : Gi−1 → Gi−1/N_i be the corresponding quotient map, andP_i be a subgroup system overGi−1/N_i that containsG_i/N_i. Define

P ={gπ_i⁻¹(H)g⁻¹ : 1≤i≤k, H ∈ P_i, g ∈G₀},

which is a subgroup system overG₀ and containsπ_i⁻¹(G_i/N_i) =G_i for alli∈[k].

Then we have

1. If for alli∈[k], all strongly antisymmetricPi-schemes are discrete onGi/Ni, then all strongly antisymmetricP-schemes are discrete onG_k.

2. If for somei∈[k], all strongly antisymmetricP_i-schemes are inhomogeneous onG_i/N_i, then all strongly antisymmetricP-schemes are inhomogeneous on G_k.

The same holds if strong antisymmetry is replaced by antisymmetry.

We defer the proof of Lemma 4.12 to Section 6.1.

Proof of Theorem 4.2. The claims aboutKi andgi follow from Theorem 4.1. Use the following notations fori∈[k]:

• L_i: the splitting field ofg_i overKi−1, which is a subfield ofL.

• G_i := Gal(L_i/Ki−1)andN_i := Gal(L/L_i).

• π_i: the natural projectionGal(L/K_i−1)→Gal(L/K_i−1)/N_i ∼=G_i.

• P_i: the subgroup system overG_i associated withF_i.

Then by construction, the subgroup system overGal(L/K₀)associated withF is P :={gπ_i⁻¹(H)g⁻¹ : 1≤i≤k, H ∈ P_i, g ∈G}.

Assume the conditions on Fi in Theorem 4.2 are satisfied. Then for all i ∈ [k], all strongly antisymmetric P_i-schemes are discrete (resp. inhomogeneous) on Gal(L_i/K_i)∈ P_i. Applying Lemma 4.12 to the chain

Gal(L/K_k)⊆Gal(L/Kk−1)⊆ · · · ⊆Gal(L/K₁)⊆Gal(L/K₀)

andN_i,π_i,P_i, we conclude that all strongly antisymmetricP-schemes are discrete (resp. inhomogeneous) on the subgroupGal(L/K_k)∈ P. And the corresponding fixed fieldK_kis isomorphic toK₀[X]/(g(X))overK₀, as desired.

The total running time of the algorithmPrimitiveActionand the total size ofF_i are both bounded byPk

i=1T(Ki−1, g_i). The other operations take time polynomial in the total size ofF_i and the size of the input. The claim about the running time follows.

As an application, we prove the main result of [Evd92] for the special case that the input polynomial satisfies Condition 3.1 (i.e., it is defined overFp, square free, and complete reducible overF^p).

Theorem 4.3([Evd92]). Under GRH, there exists a deterministic polynomial-time algorithm that, given a polynomial f(X) ∈ F^p[X]satisfying Condition 3.1 and a lifted polynomial f˜(X) ∈ Z[X] of f whose Galois group Gal( ˜f /Q) is solvable, computes the complete factorization off overFp.

The proof relies on the following bound for the orders of primitive solvable permu- tation groups, proved by Pálfy [Pál82].

Theorem 4.4([Pál82]). LetGbe a primitive solvable permutation group on a set of cardinalityn ∈N⁺. Then|G| ≤24^−1/3n^cfor a constantc= 3.24399. . .. Proof of Theorem 4.3. As in Section 6, we factorize f˜into its irreducible factors f₁(X), . . . , f_k(X)∈Z[X]overQin polynomial time using the factoring algorithm in [LLL82]. The Galois groupsGal( ˜f_i(X)/Q)are quotient groups ofGal( ˜f /Q), and hence are solvable as well. By replacing f(X)˜ with f˜_i(X) and f(X) with f_i(X) := ˜f_i(X) modp ∈ Fp[X]for each i ∈ [k], we reduce to the case that f˜is irreducible overQ.

Let L be the splitting field of f˜over Q. When Gal( ˜f /Q) acts primitively on the set of roots of f˜in L, its order is bounded by a polynomial in deg(f) by Theorem 4.4. Then by Theorem 4.9, we can constructF in polynomial time such thatQ[X]/( ˜f(X))∈ F and all strongly antisymmetricP-schemes are discrete on H, whereP is the subgroup system overGal( ˜f /Q) associated withF andH is a subgroup inP satisfying L^H ∼= Q[X]/( ˜f(X)). By Theorem 4.2, we also have a polynomial-time algorithm of constructing suchF in the general case. The theorem then follows from Theorem 3.9.

In Chapter 5, we prove a generalization of Theorem 4.3 (see Theorem 5.13), which implies the main result of [Evd92] in its general form. In particular, the assumption thatf˜satisfies Condition 3.1 is no longer required.