5. El Almacén Temporal Centralizado (ATC)
5.4. El ATC español
5.4.3. Proceso de autorización y aspectos sociales
4.4
Field degrees
We will complete the proof of Theorem 4.2 in this section. We recall the definition of the radical extensionB ofQ∗:
B=hQ∗, µd, ai√ci:i∈ {1, . . . , k}i.
Proof of proposition 4.3. SinceB is a Galois radical group overQ∗ by construction,
Theorem 2.25 shows that B — with the Galois action of Gal( ¯Q/Q) — has an entanglement group E(B). Recall that in this case, E(B) is the cokernel of the natural embedding of Gal(Q(B)/Q) into AutQ∗(B).
We find [Q(B) :Q(µd)] = [Q(B) :Q] ϕ(d) = #AutQ∗(B) ϕ(d)#E(B). Using Theorem 2.19 we conclude
[Q(B) :Q(µd)] = [B:Q∗] ϕ(d)#E(B) Y p prime B[p]6=Q∗[p] p−1 p = [B :Q ∗µ d]·[Q∗µd :Q∗] (d/2)·#E(B) = [B :Q∗µ d] #E(B) .
Proof of Theorem 4.2. The previous two sections show how to compute the quanti- ties [B : Q∗µ
d] and #E(B) in this fraction in the required time. Combining these
statements yields Algorithm 4.16 to compute [Q(B) :Q(µd)].
This algorithm runs in time polynomial in the input. The computation and factoring over the co-prime basis is polynomial time due to Theorem 4.4. Computing the intersections and equality insideM[2] is basic linear algebra overF2with matrix
sizes linear in the input size.
Finally, evaluating the expression from Theorem 4.7 involves computing an order of a Z/dZ-moduleD, where the generators of D are written on a basis of the free
Z/dZ-moduleM of rank, say,r. The computation of this order can be performed by dividing dr by the index of ((dZ)r+D) inside Zr. Since both the number of generators of D and the size of each coefficient are linear in the size of the input, this index can be computed in polynomial time, using for example the methods from [8].
Algorithm 4.16.
1. Determine a coprime baseP for the set consisting of 2, allai, and all ci and
factor all these numbers overP.
2. Computed= lcm{ai}using this factorization.
3. Define theZ/dZ-moduleM as in Lemma 4.5.
4. Use Equation 4.6 and Theorem 4.7 to compute [B :Q∗µ
d].
5. Ifdis odd, then #E(B) = 1. Proceed with step 10.
6. Use Lemma 4.11 to findF2-bases ofMd andM2d insideM[2].
7. Use Proposition 4.12 to find WGµ/Q∗ by computing (W/Q∗)∩M
2d inside
M[2].
8. Compute the intersections (WGµ∩B)/Q∗ and (W/Q∗)∩M
d insideM[2].
9. Use Corollary 4.15 to compute #E(B).
10. Finally, use Proposition 4.3 to compute [Q(B) :Q(µd)].
Example 4.17.
ConsiderK=Q(µ12,√66,√4−9), or, equivalently
K=Qµ12,
12√
62, 12p −36.
Using the notation from throughout this chapter, we getd= 12 and
B=DQ∗, µ12,
12√
62, 12p
−36E.
The coprime base for the numbers involved necessarily consists of actual primes in this case: P ={2,3}. This means that the freeZ/dZ-moduleM is given by
M =DQ∗, µ24, 12 √
2, 12√
3E .Q∗.
Inside this module, we compute the index [B : µ12Q∗]. On the (ordered)
basis (ζ24, 12 √
2, 12√
3), the submodule µ12Q∗ is generated by h(2,0,0)i andB
byh(2,0,0),(0,2,2),(1,0,6)i. Adding 3·(0,2,2) to (1,0,6) results in a basis forB in triangular form:
B =h(2,0,0),(1,6,0),(0,2,2)i.
4.4. Field degrees 61
We continue with the entanglement group computation, starting withE(B0)
forB0=hB, µ
24i. This takes place in the 2-torsion ofM:
M[2] =D√−1,√2,√3E.
Since √−1,√2 and √3 are all contained in Q(µ24), the intersection M24 =
M[2]∩(Q(µ24)∗/Q∗) actually equalsM[2]. The three roots√−1,√2 and√3
are also all contained in B0, soWGµ/Q∗ = M
24∩(W/Q∗) = M24∩B0 also
equalsM[2].
As an aside, according to Theorem 4.13 the order of E(B0) is half that of WGµ/Q∗, so we have #E(B0) = 4. To compute the size of E(B) from this,
we need to determine if (WGµ∩B)/Q∗equals (W/Q∗)∩M
12. Since√3 is not
in B, it is not in the former module, while it is in the latter, so they are not equal, and #E(B) = 8·1
4 = 2.
This leads us to the conclusion
h Qµ12, 6 √ 6,√4 −9:Qi=ϕ(12)·12 2 = 24.
Chapter 5
Near-primitive roots and
higher rank
5.1
Introduction
In this chapter we generalize the results from Chapter 1 to a broader setting. LetK be a number field, and letV ⊂K∗ be a finitely generated subgroup with
rank(V /Vtors)≥1 andta positive integer. We consider the set M =M(K, V, t) of
primes qofK satisfying:
• ordq(v) = 0 for allv∈V, and
• [(OK/q)∗: ¯V]|t.
This is a special case of the broader context considered by H.W. Lenstra [18]. If we take V to be generated by a single element, this element is called a near- primitive root modulo the primes q satisfying the conditions. Over the rationals, these densities have previously been computed; see Wagstaff [35] and Moree [23].
If on the other hand we taket = 1, butV generated by multiple elements, this leads to higher rank analogues of Artin’s conjecture. ForK=Q, this topic has been treated by Cangelmi and Pappalardi [6], and is covered in a way very similar to the approach in this chapter by Moree and Stevenhagen [24].
The work of Cooke and Weinberger [9] shows that the setM(K, V, t) has a natural density under the appropriate generalized Riemann hypotheses.
First of all, note that the set of primesqnot satisfying the first condition is finite, since V is finitely generated. After all, it is sufficient to check this condition for a set of generators ofV.
Following the same strategy as in Chapter 1, we will see that the second condition can also be translated to splitting conditions on radical extension fields of K.
Specifically, for a (rational) primep, lete(p) be the smallest positive integer such that pe(p)does not dividet, and define the radical extensions
K∗⊂Bp=hK∗, µpe(p), pe(p)√
Vi.
Here pe(p)√
V denotes the group of all elementsxin a fixed algebraic closure ¯K ofK
that satisfy xpe(p)
∈ V. Let B be the abelian group generated by all Bp, and let
E = E(B) be its entanglement group with respect to the action of the absolute Galois group of K.
Theorem 5.1. The entanglement group E=E(B)of B is finite.
AsA= AutK∗(B) is naturally isomorphic to the product of allAp= AutK∗(Bp)
and E is finite, only a finite number of Ap have a non-trivial image in E. This
ensures that the (a priori infinite) product in the correction factor formula below is in fact a finite product.
Theorem 5.2. Assuming GRH, the set M(K, V, t)has a natural density equal to
C(K, V, t)· Y pprime 1− 1 #Ap ,
whereC(K, V, t)is a rational correction factor given by
C(K, V, t) = X χ∈E∨ Y p prime χ(Ap)6=1 −1 #Ap−1.
We prove these two main theorems in the following section.
In this generality, the results from Chapter 1 no longer suffice to give explicit expressions forE andχ(Ap). In the remainder of the chapter we will address these
issues, using the theory from Chapters 2 and 4.