Other techniques of constructing number fields

Chapter IV: Constructing number fields

4.4 Other techniques of constructing number fields

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.

Taking the compositum of number fields. Note that the fields computed in the two algorithmsSplittingField and Stabilizersin Section 4.2 are (up to isomorphism overK₀) compositums of conjugates of the fieldK₀[X]/(g(X)). The general problem of constructing the compositum of (relative) number fields is solved by the following lemma.

Lemma 4.13. There exists a polynomial-time algorithm that given a number field K₀ and relative number fieldsK, LoverK₀, constructs all the compositumsK⁰L⁰ up to isomorphism over K₀ where K⁰ (resp. L⁰) ranges over the conjugates of K (resp. L) overK₀ in the algebraic closureK¯₀ ofK₀.6

Proof. Take the irreducible polynomialg(X)∈K₀[X]that encodesL, i.e.,L∼=K0

K₀[X]/(g(X)). Factorize g(X) into irreducible polynomials g₁(X), . . . , g_k(X) overK. Then compute and output the fieldsK[X]/(g₁(X)), . . . , K[X]/(g_k(X)). To see that this gives the desired output, note that we may fix K = K⁰ as fields are constructed only up to isomorphism over K₀. Letα₁, . . . , α_n be the roots ofg in K¯₀, wheren = deg(g). Then the conjugates ofLin K¯₀ overK₀ are precisely K₀(α₁), . . . , K₀(α_n). Fori∈[n], there exists a uniquej_i ∈[k]such thatα_iis the root ofg_ji, and the compositum ofKandK₀(α_i)is justK(α_i)∼=K0 K[X]/(g_ji(X)). Taking the intersection of number fields. The intersection of two number fields can be computed efficiently, as shown in [LM85].

Theorem 4.5([LM85]). There exists a polynomial-time algorithm that given

• number fieldsK = Q(α), K⁰ = Q(β)encoded by the minimal polynomials of primitive elementsα ∈K andβ∈K⁰overQrespectively, and

• the minimal polynomialh₀(X)∈K[X]ofβ overK,7 computes the number fieldK∩K⁰ up to isomorphism.

The algorithm in [LM85] also extends to relative number fields. We omit the details.

6HereK andL are embedded inK¯0 via someK0-linear embeddings. The choices of these embeddings do not matter as we constructK⁰L⁰for all the conjugatesK⁰andL⁰overK0.

7The polynomialh₀is needed for the problem to be well defined.

Adjoining a square root of the discriminant. SupposeK is a relative number field over K₀ encoded by the minimal polynomial h(X) ∈ K₀[X] of a primitive element α ∈ K over K₀. Let L be the Galois closure of K/K₀ and let G = Gal(L/K₀). ThenGacts on the setSof roots ofhinLand hence can be identified with a subgroup ofSym(S).

SupposeS ={α₁, . . . , α_n}. Define thediscriminantofhto be

∆h := Y

1≤i<j≤n

(αi−αj)².

We have^g∆_h = ∆_h for allg ∈G. So∆_h ∈L^G=K₀. Now consider the subfieldK₀⁰ :=K₀(√

∆_h)ofL, where√

∆_h :=Q

1≤i<j≤n(α_i−α_j) is a square root of∆_hinL. A permutationg ∈Gfixes√

∆_hprecisely whengis an even permutation ofS, which implies

Gal(L/K₀⁰) =G∩Alt(S).

With this observation, we have

Lemma 4.14. There exists a polynomial-time algorithm that given a number field K₀ and a relative number fieldK over K₀ encoded by h(X) ∈ K₀[X], computes L^G∩Alt(S) up to isomorphism over K₀, where L is the Galois closure of K/K₀, G= Gal(L/K₀), andSis the set of roots ofhinL.

Proof. We haveL^G∩Alt(S) = K₀(√

∆_h)by the above discussion. Letn = deg(h). Then discriminant∆_h satisfies the identity

∆h = (−1)^n(n−1)/2Res(h, h⁰),

where Res(h, h⁰) denotes the resultant of hand its derivative h⁰. and is given by the determinant of the Sylvester matrix associated with h and h⁰ [Lan02]. Thus we can compute ∆_h in polynomial time. Then we test if ∆_h is a square in K₀ by factoringX² −∆_h overK₀. If∆_h is a square, we have K₀(√

∆_h) = K₀ and correspondinglyG⊆Alt(S). In this case we just outputK₀. Otherwise we output K₀[X]/(X²−∆_h).

Remark. The technique above was used in [Lan84] for the determination of the Galois groups of number field extensions. It is not clear, however, if it helps for the problem of polynomial factoring over finite fields. We note that replacing K₀

withK₀ =K₀( ∆_h)andK withK₀Khas the effect of reducing the Galois group G toG∩Alt(S), but the order of Gis reduced by at most a factor of two. This does not help in the case that G = Sym(S) and P is a system of stabilizers of depth m ≤ |S| −2 (with respect to the natural action of G): As both Sym(S) andAlt(S)arek-transitive for k =|S| −2,P-schemes forSym(S)and those for Alt(S)both correspond tom-schemes onS(see Theorem 2.1), and hence they are the equivalent.

Computing the fixed field of the automorphism group. The following lemma gives a characterization of the fixed field of an automorphism subgroup.

Lemma 4.15. SupposeK/K₀is a field extension andαis a primitive element ofK overK₀. For a subgroupU ⊆Aut(K/K₀), the fieldK^Uis generated by elementary symmetric polynomials in the elements^gα(indexed byg ∈U) overK₀.

Proof. LetK⁰be the subfield ofKgenerated by elementary symmetric polynomials in^gα,g ∈U overK₀. We obviously haveK⁰ ⊆K^U. By Galois theory, it holds that [K : K^U] = |U|(see, e.g., [Lan02, Section VI.1, Theorem 1.8]). So it suffices to prove[K :K⁰]≤ |U|.

Consider the polynomial φ(X) = Q

g∈U(X −^gα). The coefficients of φ(X) are, up to sign, given by elementary symmetric polynomials in ^gα, g ∈ U and hence φ(X)∈K⁰[X]. Asφ(α) = 0, the minimal polynomial ofαoverK⁰ dividesφ(X), and its degree is at mostdeg(φ) =|U|. So we have[K⁰(α) :K⁰]≤ |U|. The claim follows by noting thatK⁰(α) =K.

Lemma 4.15 provides a method of computing the fixed field of the automorphism groupAut(K/K₀):

Theorem 4.6. There exists a polynomial-time algorithm that given a number field K₀and a relative number fieldKoverK₀, computes the fixed fieldK^Aut(K/K0)⊆K. Proof. SupposeK is encoded by the minimal polynomial of a primitive elementα overK₀. We compute all the automorphisms ofKinAut(K/K₀)using Lemma 4.7.

Then we adjoining to K₀ the first k elementary symmetric functions in ^gα, g ∈ Aut(K/K0)wherek =|Aut(K/K0)|. The resulting field is exactlyK^Aut(K/K0)by Lemma 4.15.

More generally, givenK₀, K and a subgroupU ⊆ Aut(K/K₀)of automorphisms ofK, the same proof shows thatK^U can be constructed in polynomial time.

Now supposeLis a Galois extension ofK₀ that containsK. LetG= Gal(L/K₀) andH = Gal(L/K). ThenAut(K/K₀)is identified withN_G(H)/H, and we have K^Aut(K/K0) = K^NG(H)/H = L^NG(H). So Theorem 4.6 states that L^NG(H) can be constructed in polynomial time givenK =L^HandK₀. In the context of polynomial factoring using theP-scheme algorithm, this means that we can efficiently enlarge a subgroup systemP by including the normalizersN_G(H)ofH ∈ P.

A natural question arising from this observation is whether adding the normalizers (or more generally subgroups betweenN_G(H)andH) to the subgroup system helps a P-scheme algorithm obtain the complete factorization (resp. a proper factorization).

By Theorem 3.9, this reduces to the question whether it helps for proving all strongly antisymmetric P-schemes are discrete (resp. inhomogeneous) on a distinguished subgroupH ∈ P.

For discreteness of strongly antisymmetricP-schemes, we give an affirmative an- swer in general: we show that for some subgroup system P and H ∈ P, there exist strongly antisymmetric P-schemes that are not discrete on H, but adding normalizers to the subgroup system rules out their existence.

Example4.1. Choose a finite groupGand a subgroupH ⊆Gsuch thatNG(H)is a proper normal subgroup ofG.8 ChooseP ={gHg⁻¹ :g ∈G}which is a subgroup system overG. Define aP-collectionC ={C_H⁰ :H⁰ ∈ P}as follows: the group N_G(H)acts on H\Gby left translation ^gHh = HghandH\Gis partitioned into N_G(H)-orbits. Choose a complete set of representativesB ⊆H\Gfor these orbits.

DefineC_H ={^gB : g ∈N_G(H)}. For any other subgroupH⁰ inP, chooseg ∈G such that H⁰ = gHg⁻¹, and define C_H⁰ = {c_H,g(B) : B ∈ C_H}. It is easy to see that C is a well defined strongly antisymmetric P-scheme. Moreover, it is not discrete onHsinceN_G(H)does not act transitively onH\G.

Now defineP⁰ =P ∪ {N_G(H)}which is also a subgroup system overG. We claim that any antisymmetricP⁰-schemes C⁰ must be discrete on any subgroup inP⁰. To see this, note thatC⁰ is discrete onNG(H)∈ P⁰sinceNG(H)is normal inG. Then C⁰is also discrete on all the other subgroupsH⁰ ∈ P⁰by compatibility, and the claim follows. In particular, it is impossible to extendC to an antisymmetricP⁰-scheme.

8For example, we may chooseGto be the semidirect product(K×K)oC2, where Kis a nontrivial finite group andC₂permutes the two direct factors ofK×K. LetH =K× {e}. Then NG(H) =K×KEG.

Despite the example above, adding normalizers to the subgroup system seems not helpful for attacking the most difficult cases in polynomial factoring: for a subgroup systemP over a finite groupG, define

P₊={U :H ⊆U ⊆N_G(H), H ∈ P},

which is also a subgroup system over G. For several important families of per- mutation groups, we show that if P is the corresponding system of stabilizers of certain depth m(wherem is not too large), anyP-scheme C can be extended to a P+-schemeC⁰with antisymmetry and strong antisymmetry preserved. In particular, ifC is not discrete or inhomogeneous on some subgroupH ∈ P, then neither isC⁰. Lemma 4.16. LetS be a finite set and letGbeSym(S)orAlt(S)acting naturally onS. LetPbe the system of stabilizers of depthmoverGwith respect to this action wherem < |S|/2. Then anyP-schemeC can be extended to aP+-schemeC⁰ such thatC⁰is antisymmetric (resp. strongly antisymmetric) if so isC.

Lemma 4.17. LetV be a finite dimensional vector space over a finite field and letG beGL(V)acting naturally onS :=V − {0}. Let P be the system of stabilizers of depthmoverGwith respect to this action wherem <dimF V. Then anyP-scheme C can be extended to aP₊-scheme C⁰ such thatC⁰ is antisymmetric (resp. strongly antisymmetric) if so isC.

We defer the proofs of Lemma 4.16 and Lemma 4.17 to Section 6.4 . There we define theclosureP_cl of a subgroup systemP, and then show that P-schemes can always be extended to P_cl-schemes with antisymmetry and strong antisymmetry preserved. Lemma 4.16 and Lemma 4.17 then follow immediately once we verify thatP_cl =P₊in these cases.

C h a p t e r 5

THE GENERALIZED P -SCHEME ALGORITHM

In Chapter 3, we developed the P-scheme algorithm that factorizes polynomials satisfying Condition 3.1, i.e., they are defined over a prime fieldF^p, square-free, and completely reducible overF^p. In this chapter, we extend this algorithm to factorize general polynomials f(X) ∈ F^q[X] over a finite fieldF^q of characteristic p. The generality is reflected in the following three aspects: (1) Fq may be a non-prime field, (2) the degrees of the irreducible factors off may be greater than one, and (3) the multiplicities of the irreducible factors off may be greater than one.

Motivation. Techniques like Berlekamp’s reduction [Ber70], square-free factor- ization [Yun76; Knu98] and distinct-degree factorization [CZ81] were commonly used in literature to reduce the problem to the special case that the input polynomial satisfies Condition 3.1. However, these reductions do not preserve the information of the lifted polynomial f˜employed by the P-scheme algorithm. Therefore, it is desirable to avoid these reductions and extend theP-scheme algorithm to the general setting instead.

As a concrete example, consider the following polynomialf(X)˜ ∈Z[X]irreducible overQ, taken from [KM00]:

f˜(X) =X¹⁴+ 28X¹¹+ 28X¹⁰−28X⁹+ 140X⁸+ 360X⁷ + 147X⁶ + 196X⁵+ 336X⁴−546X³−532X²+ 896X+ 823.

Forp= 43, the reduced polynomialf(X) = ˜f(X) mod phas seven distinct linear factors and one irreducible factor of degree 7 overFp:

f(X) =(X+ 2)(X+ 4)(X+ 9)(X+ 19)(X+ 23)(X+ 30)(X+ 42) (X⁷+ 14X⁴+ 15X³+ 31X²+ 15X+ 38).

The standard way of factoringfoverFpis first applying distinct-degree factorization [CZ81] to obtain a partial factorizationf =f0f1, where

f0(X) = (X+ 2)(X+ 4)(X+ 9)(X+ 19)(X+ 23)(X+ 30)(X+ 42) is the product of the linear factors and satisfies Condition 3.1. Then we factorize f0 over F^p. To achieve this goal deterministically, we pick a lifted polynomial

f˜₀(X)∈Z[X]off, which we may assume to be irreducible, and run theP-scheme algorithm in Chapter 3. Suppose the (Q,f)˜-subfield system in the algorithm is constructed by Lemma 3.21 and the associated subgroup systemP is the system of stabilizers of depth m, where m ∈ N⁺ is sufficiently large. In the worst case, the action ofGal( ˜f₀/Q)on the set of roots off˜is permutation isomorphic to the natural action of the symmetric group Sym(7) on[7]. Then we need m ≥ 3to obtain a proper factorization of f, since by Theorem 2.1 and Lemma 2.19, there exists a strongly antisymmetricP-scheme homogeneous on a stabilizer ifm≤2.1

On the other hand, the action of the Galois group of f˜on the set of roots of f˜is permutation isomorphic to the action of the wreath product2 C₇ oC₂ on[7]×[2], where C₇ permutes [7] cyclically and C₂ permutes the two copies of [7]. This action has a base of size two, which suggests that choosing m = 2 is sufficient for completely factoringf, provided that we have a generalization of Theorem 3.2 that employs the polynomial f˜. The goal of this chapter is to establish such a generalization.

The example above generalizes to an infinite family of instances: for everyk ∈N⁺, there existsf˜(X)∈Z[X]irreducible overQof degree2ksuch that the action of the Galois group on the set of roots off˜is permutation isomorphic to the action ofC_koC₂ on [k]×[2].3 And for such f˜, there exists infinitely many prime numberspsuch thatf(X) = ˜f(X) modphaskdistinct linear factors and one irreducible factor of degree k.4 Using the generalized P-scheme algorithm developed in this chapter, it is sufficient to choosem = 2in order to completely factorizef˜modp, leading to a polynomial-time factoring algorithm for such instances (f,f)˜. On the other hand, using distinct-degree factorization and theP-scheme algorithm in Chapter 3, the best known general upper bound formisO(logk)(see Theorem 3.12), and the resulting algorithm takes superpolynomial time.

Lifted polynomial. To formulate the main result of this chapter, we first need to generalize the notion of lifted polynomials (see Definition 1.1). Recall that a lifted polynomial of f(X) ∈ Fp[X] is a polynomial f(X)˜ ∈ Z[X] of degree

1For the same reason, one needs to choosem≥3if them-scheme algorithm [IKS09] is used.

2For the definition of the wreath product of groups, see Definition 6.11.

3Shafarevich’s theorem on solvable Galois groups [Sha54; ILF97] implies that the existence of integral polynomials realizing the family of groupsC_k oC₂ as Galois groups. For an algorithm explicitly computing such a polynomial, see [KM00].

4This follows from Chebotarëv’s density theorem. See, e.g., [Neu99].

deg(f) satisfying f˜(X) modp = f(X). For the general case F^q = Fp^d, we fix the following notations: assumeF^q is encoded by a monic irreducible polynomial h(Y) ∈ Fp[Y] of degreed, i.e., it is identified with Fp[Y]/(h(Y))via an isomor- phism ψ₀ : Fp[Y]/(h(Y)) → Fq which we can efficiently compute. Lift h to a monic polynomial ˜h(Y) ∈ Z[Y]of degree d which is necessarily irreducible over Q. DefineA₀ :=Z[Y]/(˜h(Y))andK₀ :=Q[Y]/(˜h(Y)). Composing ψ₀ with the natural projectionA₀ →Fp[Y]/(h(Y))sendingx toxmodp, we obtain a surjec- tive ring homomorphism ψ˜₀ : A₀ → Fq. Finally extend ψ˜₀ to the ring A₀[X] by applying it to each coefficient:

ψ˜₀ :A₀[X]→F^q[X].

With these notations, we generalize the definition of lifted polynomials as follows.

Definition 5.1 (lifted polynomial). Suppose f(X) ∈ F^q[X] is a polynomial of degreen ∈N⁺. Alifted polynomialoff (with respect to˜handψ₀) is a polynomial f˜(X) ∈A₀[X]of degreensatisfyingψ˜₀( ˜f) =f. Anirreducible lifted polynomial off is a lifted polynomial off that is irreducible overK₀.

Givenf(X)∈F^q[X], we can choose a lifted polynomialf˜offefficiently. Further- more, we argue that f˜can be assumed to be irreducible overK₀. To see this, we need the following lemma.

Lemma 5.1. There exists a polynomial-time algorithm that given p and a poly- nomial f(X)˜ ∈ A₀[X] satisfying ψ˜₀( ˜f) 6= 0, computes an integer D satisfying D ≡ 1 (mod p)and a factorization of D·f˜into irreducible factorsf˜_i overK₀. Furthermore all of the factorsf˜_i(X)are inA₀[X].

The proof can be found in Appendix C. ComputeDandf_i using the lemma above.

We haveψ˜₀(D·f) = ˜˜ ψ₀( ˜f) =f sinceD≡1 (mod p). So the polynomialsψ˜₀( ˜f_i) are factors off, and we have reduced the problem to factoring eachψ˜₀( ˜f_i)∈Fq[X]

using its irreducible lifted polynomialf˜_i.

The discussion above justifies the assumption that an irreducible lifted polynomial f˜off is given, with respect toh˜andψ₀. The notations˜h,ψ₀,A₀, andK₀are fixed throughout this chapter.

Main result. The main result of this chapter is a generalization of Theorem 3.2:

Theorem 5.1(informal). Suppose there exists a deterministic algorithm that given a polynomialg(X)∈A₀[X]irreducible overK₀, constructs in timeT(g)a collection F of subfields of the splitting fieldLofg overK₀ such that

• F =K₀[X]/(g(X))is inF, and

• all strongly antisymmetricP-schemes are discrete onGal(L/F)∈ P, where P is the subgroup system associated withF.

Then under GRH, there exists a deterministic algorithm that givenf(X)∈ Fq[X]

and an irreducible lifted polynomial f˜(X) ∈ A₀[X] of f, outputs the complete factorization off overF^q in time polynomial inT( ˜f)and the size of the input.

See Theorem 5.9 for the formal statement. For simplicity, here we only state the result for computing the complete factorization off. The results for computing a proper factorization are slightly more complicated to state, and we refer the reader to Section 5.10 for details.

Overview of the generalizedP-scheme algorithm

Recall that aP-scheme algorithm in Chapter 3 consists of three parts: (1) a reduction to the problem of computing an idempotent decomposition of the ringO¯_F, where F = Q[X]/( ˜f(X)), (2) computing idempotent decompositions for a collection of number fields, and (3) constructing the collection of number fields used in the previous part. The factoring algorithm in this chapter has the same structure but with some differences: we generalize the reduction in Part (1), whereF now denotes the number fieldK₀[X]/( ˜f(X)). And in Part (3), we construct a collection of relative number fields overK0 instead of ordinary number fields. The main difference is in Part (2), which we now explain.

P-schemes of double cosets. In Chapter 3, we proved that for a subfieldK of the splitting fieldLoff˜,Gthe Galois group off˜, andH= Gal(L/K), an idempotent decomposition of the ring O¯_K corresponds to a partition of the right coset space H\G. The crucial condition for this claim to hold is thatpsplits completelyin the splitting field Lof f˜, which in turn relies on the assumption thatf is square-free and completely reducible over the field of definition. In general, one can prove that an idempotent decomposition ofO¯_K corresponds to a partition of thedouble coset spaceH\G/D_Q0 instead of the right coset spaceH\G, whereD_Q0 ⊆ Gis known

as the decomposition group (of a fixed prime ideal Q0 of O_L over K₀). For the special case studied in Chapter 3, the decomposition groupD_Q0 is trivial, and hence the double coset spaceH\G/D_Q0 coincides with the right coset spaceH\G. To address the general case, we define the notion ofP-collections (resp. P-schemes) of double cosets, generalizing (ordinary)P-collections (resp. P-schemes). Various properties including (strong) antisymmetry, discreteness and homogeneity can be extended to P-schemes of double cosets. In addition, as the rings O¯_K are not necessarily semisimple in general, we replace them with the ringsRK, defined by

R_K :=

x∈O¯_K/Rad( ¯O_K) :x^p =x ,

where Rad( ¯O_K) denotes the radical of O¯_K. These rings have the advantage of being finite products ofFp, so that we can directly use the results in Chapter 3. Then we generalize the algorithm in Chapter 3 to compute a collection of idempotent decompositions of the ringsR_K so that they correspond to a strongly antisymmetric P-schemes of double cosets.

In addition, we introduce the following notations concerning partitions of double coset spaces: for every double coset HgD_Q0 ∈ H\G/D_Q0 where H ⊆ G, we associate two positive integersf(HgD_Q0)ande(HgD_Q0), called theinertia degree and the ramification indexofHgD_Q0 respectively.5 Then we say a partitionP of H\G/D_Q0 has locally constant inertia degrees (resp. ramification indices) if for every blockB inP, all the double cosets inB have the same inertia degree (resp.

ramification index). We design efficient algorithms that force the partitions in aP- collection of double cosets to have locally constant inertia degrees and ramification indices. These algorithms may be regarded as the analogues of distinct-degree factorization [CZ81] and square-free factorization [Yun76; Knu98] that factorize a polynomial according to the degrees and the multiplicities of the irreducible factors.

The discussion above is summarized by the following theorem, which generalizes Theorem 3.1 in Chapter 3.

Theorem 5.2 (informal). Under GRH, there exists a deterministic algorithm that given a poset P^] of number fields between K₀ and L corresponding to a poset P of subgroups of G, outputs idempotent decompositions of R_K for K ∈ P^]

5These names come from the fact thatf(HgDQ₀) (resp. e(HgDQ₀)) is the inertia degree (resp. ramification index) of the prime ideal ofO_LH lying overpcorresponding toHgDQ₀. See Definition 5.2 for details.