Advances in Pure Mathematics
Vol.07 No.02(2017), Article ID:73971,45 pages
10.4236/apm.2017.72008
Criteria for Three-Stage Towers of -Class Fields
Daniel C. Mayer
Naglergasse 53, 8010 Graz, Austria
Copyright © 2017 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: November 24, 2016; Accepted: February 4, 2017; Published: February 7, 2017
ABSTRACT
Let p be a prime and K be a number field with non-trivial p-class group . A crucial step in identifying the Galois group of the maximal unramified pro- extension of is to determine its two-stage approximation , that is the second derived quotient . The family of abelian type invariants of the -class groups of all unramified cyclic extensions of degree is called the index- abelianization data (IPAD) of . It is able to specify a finite batch of contestants for the second -class group of . In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD includes data on unramified abelian extensions of degree and enables sharper bounds for the order of in the case , where current im- plementations of the p-group generation algorithm fail to produce explicit contestants for , due to memory limitations. The iterated IPAD of second order contains information on non-abelian unramified extensions L/K of degree p2, or even p3, and admits the identification of the p-class tower group for various infinite series of quadratic fields with possessing a p-class field tower of exact length as a striking novelty.
Keywords:
Hilbert p-Class Field Tower, p-Class Group, p-Principalization Types, Quadratic Fields, Unramified Cyclic Cubic Field Extensions, p-Class Tower Group, Relation Rank, Metabelianization, Coclass Graphs
1. Introduction
In a previous article [1] , we provided a systematic and rigorous introduction of the concepts of abelian type invariants and iterated IPADs of higher order. These ideas were communicated together with impressive numerical appli- cations at the 29th Journées Arithmétiques in Debrecen, July 2015 [2] . The purpose and the organization of the present article, which considerably extends the computational and theoretical results in [1] [2] , is as follows.
Index-p abelianization data (IPADs) are explained in §2. Our Main Theorem on three-stage towers of 3-class fields is communicated in §3. Basic definitions concerning the Artin transfer pattern [1] [3] [4] are recalled in §4. Then we generally put and consider 3-class tower groups (§7). In §5, we first restate a summary of all possible IPADs of a number field with 3-class group of type (3,3) [ [1] , Thm. 3.1-3.2, pp. 290-291] in a more succinct and elegant form avoiding infinitely many exceptions, and emphasizing the role of two distinguished components, called the polarization and co-polarization, which are crucial for proving the finiteness of the batch of contestants for the second 3-class group . Up to now, this is the unique situation where all IPADs can be given in a complete form, except for the simple case of a number field with 2-class group of type (2,2) [ [5] , § 9, pp. 501-503]. We characterize all relevant finite 3-groups by IPADs of first and second order in §§7.1, 7.3, 7.6, 7.9. These groups constitute the candidates for 3-class tower groups of quadratic fields with 3-class group of type (3,3). In §7.2, results for the dominant scenario with 3-principalization of type a are given. In §§7.5, 7.8, we provide evidence of unexpected phenomena revealed by real quadratic fields with types in Scholz and Taussky’s section [ [6] , p. 36]. Their 3-class tower can be of length and a sharp decision is possible by means of iterated IPADs of second order. We point out that imaginary quadratic fields with type E must always have a tower of exact length [3] [7] . In §§7.10, 7.11, resp. §§7.12, 7.13, results for quadratic fields with 3-principalization type , , resp. , , are proved.
In the last section §8 on multi-layered IPADs, it is our endeavour to point out that the rate of growth of successive derived quotients , , of the -class tower group is still far from being known for imaginary quadratic fields with -class rank , where the criterion of Koch and Venkov [8] ensures an infinite -class tower with .
2. Index- Abelianization Data
Let be a prime number. According to the Artin reciprocity law of class field theory [9] , the unramified cyclic extensions of relative degree of a number field with non-trivial -class group are in a bijective correspondence to the subgroups of index in . Their number is given
by if denotes the -class rank of [ [10] , Thm. 3.1]. The reason
for this fact is that the Galois group of the maximal unramified abelian -extension , which is called the first Hilbert - class field of , is isomorphic to the -class group . The fields are contained in and each group is of index in .
It was also Artin’s idea [11] to leave the abelian setting of class field theory and to consider the second Hilbert -class field , that is the maximal unramified metabelian -extension of , and its Galois group , the so-called second -class group of [ [5] [6] , p. 41], for proving the principal ideal theorem that becomes trivial when it is extended to [12] . Since is a non- decreasing tower of normal extensions for any assigned unramified abelian - extension , the -class group of , , is isomorphic to the ab- elianization of the subgroup of the second -class group which corresponds to and whose commutator sub- group is given by .
In particular, the structure of the -class groups of all unramified cyclic extensions of relative degree can be interpreted as the abelian type invariants of all abelianizations of subgroups of index in the second -class group , which has been dubbed the index- abelianization data, briefly IPAD, of by Boston, Bush, and Hajir [13] . This kind of information would have been incomputable and thus useless about twenty years ago. However, with the availability of computational algebra systems like PARI/GP [14] and MAGMA [15] [16] [17] it became possible to compute the class groups , collect their structures in the IPAD , reinterpret them as abelian quotient invariants of subgroups of , and to use this information for characterizing a batch of finitely many -groups, occasionally even a unique -group, as contestants for the second -class group of , which in turn is a two-stage approxi- mation of the (potentially infinite) pro- group of the maximal unramified pro- extension of , that is its Hilbert - class tower.
As we proved in the main theorem of [ [4] , Thm. 5.4], the IPAD is usually unable to permit a decision about the length of the -class tower of when non-metabelian candidates for exist. For solving such problems, iterated IPADs of second order are required.
3. The -Principalization Type
Until very recently, the length of the -class tower
over a quadratic field with -class rank , that is, with - class group of type , , was an open problem. Apart from the proven impossibility of an abelian tower with [ [5] , Thm. 4.1.(1)], it was unknown which values can occur and whether is possible or not. In contrast, it is known that for any number field with - class rank , i.e., with non-trivial cyclic -class group , and that for an imaginary quadratic field with -class rank , when is odd [8] .
The finite batch of contestants for , specified by the IPAD , can be narrowed down further if the -principalization type of is known. That is the family of all kernels of -class transfers from to unramified cyclic superfields of degree over . In view of the open problem for the length of the -class tower, there arose the question whether each possible -principalization type of a quadratic field with of type is associated with a fixed value of the tower length .
For and of type (3,3), there exist 23 distinct 3-principalization types [ [18] , Tbl. 6-7], designated by , where denotes a letter in and denotes a certain integer in , more explicitly:
A.1, D.5, D.10, E.6, E.8, E.9, E.14, F.7, F.11, F.12, F.13, G.16, G.19, H.4,
a.1, a.2, a.3, b.10, c.18, c.21, d.23, d.25.
In this article, we establish the last but one step for the proof of the following solution to the open problem for and quadratic fields with .
Theorem 3.1. (Main theorem on the length of the 3-class tower for 3-class rank two)
1) For each of the 13 types of 3-principalization with upper case letter , there exists an imaginary quadratic field , , of that type such that .
2) For each of the 22 types of 3-principalization , there exists a real quadratic field , , of that type such that .
Remark 3.1. Type must be excluded for quadratic base fields , according to [ [5] , Cor. 4.2]. It occurs, however, with for cyclic cubic fields with two primes dividing the conductor [19] .
Concerning the steps for the proof, we provide information in the form of Table 1. An asterisk indicates the present paper. The last step has been com- pleted in collaboration with M. F. Newman but has not been published yet [20] . Only the types G.16 and G.19 must be distinguished by their integer identifier, otherwise the types denoted by the same letter behave completely similar. Additionally, we give the smallest logarithmic order .
Remark 3.2. None of the types sets in with a length . Type D behaves completely rigid with , fixed class 3, and coclass 2. Type a is also confined to but admits unbounded nilpotency class with fixed coclass 1. For type E, we have with unbounded class and coclass for imaginary fields, and the unique exact dichotomy for real fields. For type , the length is fixed with unbounded class and coclass for real fields. The most extensive flexibility is revealed by fields of the types and , where
Table 1. Steps of the proof with references.
any finite unbounded length can occur with variable class and coclass. We expect that an actually infinite tower with is impossible for .
4. The Artin Transfer Pattern
Let be a prime number and be a pro- group with finite abelianization , more precisely, assume that the commutator subgroup is of index with an integer exponent .
Definition 4.1. For each integer , let be the nth layer of normal subgroups of containing .
Definition 4.2. For any intermediate group , we denote by the Artin transfer homomorphism from to [ [4] , Dfn. 3.1], and by the induced transfer.
1) Let be the multi-layered transfer target type (TTT)
of , where for each .
2) Let be the multi-layered transfer kernel type (TKT)
of , where for each .
Definition 4.3. The pair is called the (restricted) Artin pattern of .
Definition 4.4. The first order approximation of the TTT, resp. of the TKT, is called the index- abelianization data (IPAD), resp. index- obstruction data (IPOD), of .
Definition 4.5. is called iterated IPAD of
order of .
Remark 4.1. For the complete Artin pattern see [ [4] , Dfn. 5.3].
1) Since the 0th layer (top layer), , consists of the group alone, and is the natural projection onto the commutator quotient with kernel , we usually omit the trivial top layer and identify the IPOD with the first layer of the TKT.
2) In the case of an elementary abelianization of rank two, , we also identify the TKT with its first layer , since the 2nd layer (bottom layer), , consists of the commutator subgroup alone, and the kernel of is always total, that is , according to the principal ideal theorem [12] .
5. All Possible IPADs of 3-Groups of Type (3,3)
Since the abelian type invariants of certain IPAD components of an assigned 3-group depend on the parity of the nilpotency class or coclass , a more economic notation, which avoids the tedious distinction of the cases odd or even, is provided by the following definition [ [24] , §3].
Definition 5.1. For an integer , the nearly homocyclic abelian 3-group of order is defined by its type invariants , where the quotient and the remainder are determined un- iquely by the Euclidean division . Two degenerate cases are included by putting the cyclic group of order 3, and the trivial group of order 1.
In the following theorem and in the whole remainder of the article, we use the identifiers of finite 3-groups up to order 38 as they are defined in the SmallGroups Library [25] [26] . They are of the shape
Theorem 5.1. (Complete classification of all IPADs with [24] ) Let be a pro-3 group with abelianization of type (3,3) and metabe- lianization of nilpotency class , defect , and coclass . Assume that does not belong to the finitely many exceptions in the list below. Then the IPAD of in terms of nearly homocyclic abelian 3-groups is given by
(5.1)
where the polarized first component of depends on the class and defect , the co-polarized second component increases with the coclass , and the third and fourth component are completely stable for but depend on the coclass tree containing for in the following manner
(5.2)
Anomalies of finitely many, precisely 13, exceptional groups are summarized in the following list.
(5.3)
The polarization and the co-polarization we had in our mind when we spoke about a bi-polarization in [ [29] , Dfn. 3.2, p. 430]. Meanwhile, we have provided yet another proof for the existence of stable and polarized IPAD components with the aid of a natural partial order on the Artin transfer patterns distributed over a descendant tree [ [4] , Thm. 6.1-6.2].
Proof. Equations (5.1) and (5.2) are a succinct form of information which summarizes all statements about the first TTT layer in the formulas (19), (20) and (22) of [ [1] , Thm. 3.2, p. 291] omitting the claims on the second TTT layer . Here we do not need the restrictions arising from lower bounds for the nilpotency class in the cited theorem, since the remaining cases for small values of can be taken from [ [1] , Thm. 3.1, p. 290], with the exception of the following 13 anomalies in formula (5.3):
The abelian group , the extra special group , and the group do not fit into the general rules for 3-groups of coclass 1. These three groups appear in the top region of the tree diagram in the Figure 1 and Figure 2.
The four sporadic groups with and the six sporadic groups with do not belong to any coclass-2 tree, as shown in Figure 5, whence the conditions in Equation (5.2) cannot be applied to them.
On the other hand, there is no need to list the groups and in formula (14), the groups with in formula (15), and the groups with in formula (16) of [ [1] , Thm. 3.1, p. 290], since they perfectly fit into the general pattern. □
Remark 5.1. The reason why we exclude the second TTT layer from Theorem 5.1, while it is part of [ [1] , Thm. 3.1-3.2, pp. 290-291], is that we want to reduce the exceptions of the general pattern to a finite list, whereas the irregular case of the abelian quotient invariants of the commutator subgroup , which forms the single component of , occurs for each even value of
Figure 1. Distribution of absolute frequencies of on the coclass tree .
the coclass and thus infinitely often.
Theorem 5.2. (Finiteness of the batch of contestants for the second -class group ) If , , and denotes an assigned family of four abelian type invariants, then the set of all (isomorphism classes of) finite metabelian -groups such that
Figure 2. Distribution of minimal discriminants for on the coclass tree .
and is finite.
Proof. We have , when is malformed [ [1] , Dfn. 5.1, p. 294]. For and , Theorem 5.1 ensures the validity of the following general Polarization Principle: There exist a few components of a non-malformed family which determine the nilpotency class and the coclass of a finite metabelian -group with . Together with the Coclass Theorems [ [30] , §5, p. 164, and Equation (10), p. 168], the polarization principle proves the claim. □
6. Tables and Figures of Possible 3-Groups and
6.1. Tables
In this article, we shall frequently deal with finite 3-groups of huge orders for which no identifiers are available in the SmallGroups database [25] [26] . For instance in Table 6, and in the Figure 6 and Figure 7. A work-around for these cases is provided by the relative identifiers of the ANUPQ (Australian National University -Quotient) package [31] which is implemented in our licence of the computational algebra system MAGMA [15] [16] [17] and in the open source system GAP [32] .
Definition 6.1. Let be a prime number and be a finite -group with nuclear rank [ [30] , Equation (28), p. 178] and immediate descendant numbers [ [30] , Equation (34), p. 180]. Then we denote the ith immediate descendant of step size of by the symbol
(6.1)
for each and .
Recall that a group with nuclear rank is a terminal leaf without any descendants.
All numerical results in this article have been computed by means of the computational algebra system MAGMA [15] [16] [17] . The -group gene- ration algorithm by Newman [27] and O’Brien [28] was used for the recursive construction of descendant trees of finite -groups . The tree root (starting group) was taken to be for Table 2 and the Figure 2, Figure 1, Figure 5, for Table 3 and Figure 3, for Table 4 and Figure 4, for Table 5 and Figure 6, and for Table 6 and Figure 7. For computing group theoretic invariants of each tree vertex , we implemented the Artin transfers from a finite -group of type to its maximal subgroups in a MAGMA program script as described in [ [4] , §4.1].
6.2. Figures
Basic definitions, facts, and notation concerning descendant trees of finite - groups are summarized briefly in [ [29] , §2, pp. 410-411], [33] . They are discussed thoroughly in the broadest detail in the initial sections of [30] . Trees are crucial for the recent theory of -class field towers [34] [35] [36] , in particular for describing the mutual location of and .
Generally, the vertices of coclass trees in the Figures 1-4, of the sporadic part of a coclass graph in Figure 5, and of the descendant trees in the Figure 6 and Figure 7 represent isomorphism classes of finite 3-groups. Two vertices are connected by a directed edge if is isomorphic to the last lower
Table 2. IPOD and iterated IPAD of 3-groups of coclass .
central quotient , where denotes the nilpotency class of , and either , that is, is cyclic of order 3, or , that is, is bicyclic of type (3,3). See also [ [29] , §2.2, p. 410-411] and [ [30] , §4, p. 163-164].
The vertices of the tree diagrams in Figure 1 and Figure 2 are classified by using various symbols:
1) big full discs represent metabelian groups with defect ,
2) small full discs represent metabelian groups with defect .
In the Figures 3-5,
1) big full discs represent metabelian groups with bicyclic centre of type (3,3) and defect [ [29] , §3.3.2, p. 429],
2) small full discs represent metabelian groups with cyclic centre of order 3 and defect ,
3) small contour squares represent non-metabelian groups .
1) big contour squares represent groups with relation rank ,
2) small contour squares represent groups with relation rank .
A symbol adjacent to a vertex denotes the multiplicity of a batch of siblings, that is, immediate descendants sharing a common parent. The groups of particular importance are labelled by a number in angles, which is the identifier in the SmallGroups Library [25] [26] of GAP [32] and MAGMA [17] . We omit the orders, which are given on the left hand scale. The IPOD [ [18] , Thm. 2.5, Tbl. 6-7], in the bottom rectangle concerns all vertices located vertically above. The first, resp. second, component , resp. , of the IPAD [ [1] , Dfn. 3.3, p. 288] in the left rectangle concerns vertices on the same horizontal level with defect . The periodicity with length 2 of branches, for , resp. , sets in with branch , resp. , having a root of order 34, resp. 37, in Figure 1 and Figure 2, resp. 3 and 4. The metabelian skeletons of the Figure 3 and Figure 4 were drawn by
Table 3. IPOD and iterated IPAD of 3-groups on .
Nebelung [ [37] , p. 189 ff], the complete trees were given by Ascione and coworkers [38] , [ [39] , Fig. 4.8, p. 76, and Fig. 6.1, p. 123].
We define two kinds of arithmetically structured graphs of finite - groups by mapping each vertex of the graph to statistical number theo- retic information, e.g. the distribution of second -class groups or -class tower groups , with respect to a given kind of number fields , for instance real quadratic fields with discriminant
Table 4. IPOD and iterated IPAD of 3-groups on .
.
Definition 6.2. Let be a prime and be a subgraph of a descendant tree of finite -groups.
・ The mapping
(6.2)
is called the distribution of minimal discriminants on .
・ For an assigned upper bound , the mapping
(6.3)
is called the distribution of absolute frequencies on .
For both mappings, the subset of the graph consisting of vertices with
Table 5. IPOD and iterated IPAD of sporadic 3-groups G of type H.4.
, resp. , is called the support of the distribution. The trivial values outside of the support will be ignored in the sequel.
Whereas Figure 1 displays an -distribution, the Figures 2-4 show - distributions. The Figures 5-7 contain both distributions simultaneously.
7. 3-Class Towers of Quadratic Fields and Iterated IPADs of Second Order
7.1. 3-Groups of Coclass
Table 2 shows the designation of the transfer kernel type [37] , the IPOD , and the iterated multi-layered IPAD of second order,
(7.1)
for 3-groups G of maximal class up to order , characterized by the logari- thmic order, , i.e. , and the SmallGroup identifier, id. [25] [26] .
Table 6. IPOD and iterated IPAD of sporadic 3-groups of type G.19.
Figure 3. Distribution of minimal discriminants for on the coclass tree .
The groups in Table 2 are represented by vertices of the tree diagrams in Figure 1 and Figure 2.
Figure 4. Distribution of minimal discriminants for on the coclass tree .
Figure 5. Distribution of second 3-class groups on the sporadic graph .
7.2. Real Quadratic Fields of Types a.1, a.2 and a.3
Sound numerical investigations of real quadratic fields with fundamental discriminant started in 1982, when Heider and Schmithals [40] showed the first examples of a Galois cohomology structure of Moser’s type on unit groups of unramified cyclic cubic extensions, which are dihedral of degree 6 over [ [5] , Prop. 4.2, p. 482], and of IPODs with type a.1 ( ), type a.2 ( ), and type a.3 ( ), in the
Figure 6. Distribution of 3-class tower groups on the descendant tree .
Figure 7. Distribution of 3-class tower groups on the descendant tree .
notation of Nebelung [37] . See Figure 2.
Our extension in 1991 [41] merely produced further examples for these occurrences of type a. In the 15 years from 1991 to 2006 we consequently were convinced that this type with at least three total 3-principalizations is the only possible type of real quadratic fields.
The absolute frequencies in [ [5] , Tbl. 2, p. 496] and [ [24] , Tbl. 6.1, p. 451], which should be corrected by the Corrigenda in the Appendix, and the extended statistics in Figure 1 underpin the striking dominance of type a. The distribution of the second 3-class groups with the smallest order 34, resp. 36, alone reaches for the accumulated types a.2 and a.3 together, resp. for type a.1.
So it is not astonishing that the first exception without any total 3-principalizations did not show up earlier than in 2006 [ [5] , Tbl. 4, p. 498], [ [24] , Tbl. 6.3, p. 452]. See Figure 5.
The most extensive computation of data concerning unramified cyclic cubic extensions of the 481,756 real quadratic fields with dis- criminant and 3-class rank has been achieved by M. R. Bush in 2015 [42] . In the following, we focus on the partial results for 3-class groups of type (3,3), since they extend our own results of 2010 [5] [24] .
Proposition 7.1. (IPADs of fields with type a up to [42] )
In the range 0 < d < 109 with 415,698 fundamental discriminants d of real qua- dratic fields having 3-class group of type (3,3), there exist precisely
cases ( ) with IPAD ,
cases ( ) with IPAD ,
cases ( ) with IPAD , and
cases ( ) with IPAD .
Proof. The results were computed with PARI/GP [14] , double-checked with MAGMA [17] , and kindly communicated to us by M. R. Bush, privately [42] . □
For establishing the connection between IPADs and IPODs we need the following bridge.
Corollary 7.1. (Associated IPODs of fields with type a)
1) A real quadratic field with IPAD has IPOD
either of type a.2 or of type a.3.
2) A real quadratic field with IPAD has IPOD
of type a.3, more precisely a.3*, in view of the exceptional IPAD.
3) A real quadratic field with IPAD has IPOD
of type a.1.
4) A real quadratic field with IPAD has IPOD
either of type a.2 or of type a.3.
Proof. Here, we again make use of the selection rule [ [29] , Thm. 3.5, p. 420] that only every other branch of the tree is admissible for second 3-class groups of (real) quadratic fields .
According to Table 2, three (isomorphism classes of) groups share the common IPAD , namely , whereas the IPAD unambiguously leads to the group with IPOD .
In Theorem 7.4 we shall show that the mainline group cannot occur as the second 3-class group of a real quadratic field. Among the remaining two possible groups, has IPOD and has IPOD .
The IPAD leads to three groups with
IPOD and defect of commutativity [ [29] , §3.1.1, p. 412].
Concerning the IPAD , Table 2 yields four groups
with SmallGroup identifiers . The mainline group is discouraged by Theorem 7.4, has IPOD , and the two groups have IPOD .
By the Artin reciprocity law [9] [11] , the Artin pattern of the field coincides with the Artin pattern of its second 3-class group . □
Remark 7.1. The huge statistical ensembles underlying the computations of Bush [42] admit a prediction of sound and reliable tendencies in the population of the ground state. If we compare the smaller range in [5] with the extended range in [42] , then we have a decrease
by for the union of types a.2 and
a.3,
and increases
by for type a.3*, and
by for type a.1.
Of course, the accumulation of all types a.2, a.3, and a.3* with absolute frequencies
, resp. , shows a resultant de- crease
by .
For the union of the first excited states of types a.2 and a.3, we have a stagnation
at the same percentage.
Unfortunately, the exact absolute frequency of the ground state of type a.2, resp. type a.3, is unknown for the extended range . It could be computed using Theorem 7.1. However, meanwhile we succeeded in separating all states of type a.2 and type a.3 up to by immediately figuring out the 3- principalization type with MAGMA V2.22-1. In [21] , we compare the results of this most recent tour de force of computing with asymptotic densities predicted by Boston, Bush and Hajir (communicated privately and yet unpublished, similar to [13] ).
Figure 1 visualizes 3-groups of section §7.1 which arise as 3-class tower groups of real quadratic fields , , with princi- palization types a.2 and a.3 and the corresponding absolute frequencies and percentages (relative frequencies with respect to the total number of 415,698 real quadratic fields with discriminants in the range for ) which were given in Proposition 7.1.
Figure 2 visualizes 3-groups of section §7.1 which arise as 3-class tower groups of real quadratic fields , , with 3- principalization types a.1, a.2 and a.3 and the corresponding minimal dis- criminants in the sense of Definition 6.2.
As mentioned in [1] , we have the following criterion for distinguishing subtypes of type a:
Theorem 7.1. (The ground state of type a [ [29] , §3.2.5, pp. 423-424])
The second 3-class groups with the smallest order 34 possessing type a.2 or a.3 can be separated by means of the iterated IPAD of second order
.
Proof. This is essentially [ [1] , Thm. 6.1, p. 296] but can also be seen directly by comparing the column with the IPAD for the rows with and in Table 2. Here the column , containing the second layer of the IPAD, does not permit a distinction. □
Unfortunately, we also must state a negative result:
Theorem 7.2. (Excited states of type a [ [29] , §3.2.5, pp. 423-424])
Even the multi-layered IPAD of
second order is unable to separate the second 3-class groups with order 36 and type a.2 or a.3. It is also unable to distinguish between the three candidates for of type a.1, and between the two candidates for of type a.3, both for orders .
Proof. This is a consequence of comparing both columns and for the rows with and , resp. in Table 2. According to the selection rule [ [29] , Thm. 3.5, p. 420], only every other branch of the tree is admissible for second 3-class groups of (real) quadratic fields . □
Theorem 7.3. (Two-stage 3-class towers of type a) For each (real) quadratic field with second 3-class group of maximal class the 3-class tower has exact length .
Proof. Let be a 3-group of maximal class. Then is metabelian by [ [29] , Thm. 3.7, proof, p. 421] or directly by [ [43] , Thm. 6, p. 26]. Suppose that is a non-metabelian 3-group of derived length such that . According to [ [4] , Thm. 5.4], the Artin patterns and coincide, in particular, both groups share a common IPOD , which contains at least three total kernels, indicated by zeros, [18] . However, this is a contradiction already, since any non-metabelian 3-group, which necessarily must be of coclass at least 2, is descendant of one of the five groups with whose IPODs possess at most two total kernels, and a descendant cannot have an IPOD with more total kernels than its parent, by [ [4] , Thm. 5.2]. Consequently, the cover of in the sense of [ [23] , Dfn. 5.1] consists of the single element .
Finally, we apply this result to class field theory: Since is assumed to be of coclass , we obtain and the length of the 3-class tower is given by . □
Remark 7.2. To the very best of our knowledge, Theorem 7.3 does not appear in the literature, although we are convinced that it is well known to experts, since it can also be proved purely group theoretically with the aid of a theorem by Blackburn [ [43] , Thm. 4, p. 26]. Here we prefer to give a new proof which uses the structure of descendant trees.
Theorem 7.4. (The forbidden mainline of coclass 1) The mainline vertices of the coclass-1 tree cannot occur as second 3-class groups of (real) quadratic fields (of type a.1).
Proof. Since periodicity sets in with branch in the Figure 1 and Figure 2, and MAGMA shows that the groups and have -multi- plicator rank 4, all mainline vertices must have -multiplicator rank and thus relation rank . However, a real quadratic field has torsion free Dirichlet unit rank and certainly does not contain the (complex) primitive third roots of unity. According to the corrected version [ [23] , Thm. 5.1] of the Shafarevich theorem [44] , the relation rank of the 3-tower group , which coincides with the second 3-class group by Theorem 7.3, is bounded by , where denotes the 3-class rank of . □
7.3. 3-Groups G of Coclass Arising from
Table 3 shows the designation of the transfer kernel type [6] [37] , the IPOD , and the iterated multi-layered IPAD of 2nd order,
for 3-groups on the coclass tree up to order , characte- rized by the logarithmic order, , and the SmallGroup identifier, , [25] [26] . To enable a brief reference for relative identifiers we put , since this group was called the non-CF group by Ascione [38] [39] .
The groups in Table 3 are represented by vertices of the tree diagram in Figure 3.
Theorem 7.5. (Smallest possible 3-tower groups of type E.6 or E.14 [1] ). Let be a finite 3-group with IPAD of first order , where and is given in ordered form.
If the IPOD of is of type E.6, , resp. E.14, , then the IPAD of second order , where the maximal subgroups of index 3 in are denoted by , determines the isomorphism type of in the following way:
1) if and only if for if and only if , resp. or ,
2) if and only if for if and only if , resp. or ,
whereas the component is fixed and does not admit a distinction.
Proof. This is essentially [ [1] , Thm. 6.2, pp. 297-298]. It is also an immediate consequence of Table 3, which has been computed with MAGMA [17] . As a termination criterion we can now use the more precise [ [4] , Thm. 5.1] instead of [ [7] , Cor. 3.0.1, p. 771]. □
Figure 3 visualizes 3-groups which arise as second 3-class groups of real quadratic fields , , with 3-principalization types E.6 and E.14 in section §7.3 and the corresponding minimal discriminants.
7.4. Parametrized IPADs of Second Order for the Coclass Tree
Let be a descendant of coclass of the root . Denote by the nilpotency class of , by the indicator of a three-stage group, and by , resp , the defect of commutativity of itself if , and of the metabelian parent if .
Theorem 7.6. In dependence on the parameters , and , the IPAD of second order of has the form
(7.2)
where a variant of the nearly homocyclic abelian 3-group of order in Definition 5.1, which can also be defined by , , and
(7.3)
is given by and
(7.4)
7.5. Number Fields with IPOD of Type E.6 or E.14
Let be a number field with 3-class group and first layer of unramified abelian extensions.
Theorem 7.7. (Criteria for .) Let the IPOD of be of type E.6, , resp. E.14, . If with , then
・
for ,
・
for .
Proof. Exemplarily, we conduct the proof for , which is the most important situation for our computational applications.
Searching for the Artin pattern with and , resp. , in the descendant tree with abelian root , unambiguously leads to the unique metabelian descendant
with path for type E.6, resp. two descendants for type E.14. The bifurcation at the vertex with nuclear rank two leads to a unique non-metabelian descendant with path for type E.6, resp. two descendants for type E.14 The cover of is
non-trivial but very simple, since it contains two elements only. The decision whether and or and requires the iterated IPADs of second order of and , which are listed in Table 3. The general form of the component of which corresponds to the commutator subgroup is a consequence of [ [24] , Thm. 8.8, p.461], since in terms of the nilpotency class and coclass of we have and . □
The proof of Theorem 7.7, immediately justifies the following conclusions for .
Corollary 7.2. Under the assumptions of Theorem 7.7, the second and third 3-class groups of are given by their SmallGroups identifier [25] [26] , if . Independently of ,
if , then for type E.6, resp. for type E.14, and
if , then for type E.6, resp. for type E.14.
In the case of a 3-class tower of length ,
if , then for type E.6, resp. for type E.14, and
if , then for type E.6, resp. for type E.14.
The range of fundamental discriminants of real quadratic fields of type E, which underlies Theorem 7.8 in this section, resp. 7.12 in the next section, is just sufficient to prove that each of the possible groups in Theorem 7.5, resp. 7.9, is actually realized by the 3-tower group of some field .
Proposition 7.2. (Fields with IPOD of type E.6 or E.14 for [5] [24] .) In the range of fundamental discriminants of real quadratic fields , there exist precisely 3, resp. 4, cases with 3-principalization type E.6, , resp. E.14, .
Proof. The results of [ [24] , Tbl. 6.5, p. 452], where the entry in the last column freq. should be 28 instead of 29 in the first row and 4 instead of 3 in the fourth row, were computed in 2010 by means of the free number theoretic computer algebra system PARI/GP [14] using an implementation of our own princi- palization algorithm in a PARI script, as described in detail in [ [24] , §5, pp. 446-450]. The accumulated frequency 7 for the second and third row was recently split into 3 and 4 with the aid of the computational algebra system MAGMA [17] . See also [ [5] , Tbl. 4, p. 498]. □
Remark 7.3. The minimal discriminant of real quadratic fields of type E.6, resp. of type E.14, is indicated in boldface font adjacent to an oval surrounding the vertex, resp. batch of two vertices, which represents the associated second 3-class group , on the branch of the coclass tree in Figure 3.
Theorem 7.8. (3-Class towers with IPOD of type E.6 or E.14 for ) Among the 3 real quadratic fields with IPOD of type E.6 in Proposition 7.2,
・ the 2 fields ( ) with discriminants
have the unique 3-class tower group and 3-tower length ,
・ the single field ( ) with discriminant
has the unique 3-class tower group and 3-tower length .
Among the 4 real quadratic fields with IPOD of type E.14 in
Proposition 7.2,
・ the 3 fields ( ) with discriminants
have 3-class tower group or and 3-tower length ,
・ the single field ( ) with discriminant
has 3-class tower group or and 3-tower length .
Proof. Since all these real quadratic fields have 3-capitulation
type or and IPAD , and the 4 fields with have IPAD
whereas the 3 fields with have IPAD
the claim is a consequence of Theorem 7.5. □
Remark 7.4. The computation of the 3-principalization type E.14 of the field with resisted all attempts with MAGMA versions up to V2.21-7. Due to essential improvements in the change from relative to absolute number fields, made by the staff of the Computational Algebra Group at the University of Sydney, it actually became feasible to figure it out with V2.21-8 [17] for UNIX/LINUX machines or V2.22-3 for any operating system.
7.6. 3-Groups G of Coclass Arising from
Table 4 shows the designation of the transfer kernel type, the IPOD , and the iterated multi-layered IPAD of second order,
for 3-groups on the coclass tree up to order , characte- rized by the logarithmic order, , and the SmallGroup identifier, [25] [26] . To enable a brief reference for relative identifiers we put , since this group was called the non-CF group by Ascione [38] [39] .
The groups in Table 4 are represented by vertices of the tree diagram in Figure 4.
Theorem 7.9. (Smallest possible 3-tower groups of type E.8 or E.9 [1] ) Let be a finite 3-group with IPAD of first order , where and is given in ordered form.
If the IPOD of is of type E.8, , resp. E.9, , then the IPAD of second order , where the maximal subgroups of index 3 in are denoted by , determines the isomorphism type of in the following way:
1) for
if and only if , resp. or ,
2) for
if and only if , resp. or ,
whereas the component is fixed and does not admit a distinction.
Proof. This is essentially [ [1] , Thm. 6.3, pp. 298-299]. It is also an immediate consequence of Table 4, which has been computed with MAGMA [17] . As a termination criterion we can now use the more precise [ [4] , Thm. 5.1] instead of [ [7] , Cor. 3.0.1, p. 771]. □
Figure 4 visualizes 3-groups which arise as second 3-class groups of real quadratic fields , , with 3-principalization types E.8 and E.9 in section §7.6 and the corresponding minimal discriminants.
7.7. Parametrized IPADs of Second Order for the Coclass Tree
Let be a descendant of coclass of the root . Denote by the nilpotency class of , by the indicator of a three-stage group, and by , resp , the defect of commutativity of itself if , and of the metabelian parent if .
Theorem 7.10. In dependence on the parameters c, t and k, the IPAD of second order of G has the form
(7.5)
where a variant of the nearly homocyclic abelian 3-group of order is defined as in Formula (7.4) of Theorem 7.6.
7.8. Number Fields with IPOD of Type E.8 or E.9
Let be a number field with 3-class group and first layer of unramified abelian extensions.
Theorem 7.11. (Criteria for .) Let the IPOD of be of type E.8, , resp. E.9, . If with , then
・ for ,
・ for .
Proof. Exemplarily, we conduct the proof for , which is the most important situation for our computational applications.
Searching for the Artin pattern with and , resp. , in the descendant tree with abelian root , unambiguously leads to the unique metabelian descendant with path for type E.8, resp. two descendants for type E.9. The bifurcation at the vertex with nuclear rank two leads to a unique non-metabelian descendant with path for type E.8, resp. two descendants for type E.9. The cover of is non-trivial but very simple, since it contains two elements only. The decision whether and or and requires the iterated IPADs of second order of and , which are listed in Table 4. The general form of the component of which corresponds to the commutator subgroup is a consequence of [ [24] , Thm. 8.8, p. 461], since in terms of the nilpotency class and coclass of we have and .
The proof of Theorem 7.11, immediately justifies the following conclusions for .
Corollary 7.3. Under the assumptions of Theorem 7.11, the second and third 3-class groups of are given by their SmallGroups identifier [25] [26] , if . Independently of ,
if , then for type E.8, resp. for type E.9, and
if , then for type E.8, resp. for type E.9.
In the case of a 3-class tower of length ,
if , then for type E.8, resp. for type E.9, and
if , then for type E.8, resp. for type E.9.
Proposition 7.3. (Fields with IPOD of type E.8 or E.9 for [5] [24] ) In the range of fundamental discriminants of real quadratic fields , there exist precisely 3, resp. 11, cases with 3-principalization type E.8, , resp. E.9, .
Proof. The results of [ [24] , Tbl. 6.7, p. 453] were computed in 2010 by means of PARI/GP [14] using an implementation of our principalization algorithm, as described in [ [24] , 5, pp. 446-450]. The accumulated frequency 14 in the last column freq.for the second and third row was recently split into 3 and 11 with the aid of MAGMA [17] . See also [ [5] , Tbl. 4, p. 498]. □
Remark 7.5. The minimal discriminant of real quadratic fields of type E.8, resp. of type E.9, is indicated in boldface font adjacent to an oval surrounding the vertex, resp. batch of two vertices, which represents the associated second 3-class group , on the branch of the coclass tree in Figure 4.
Theorem 7.12. (3-Class towers with IPOD of type E.8 or E.9 for ) Among the 3 real quadratic fields with IPOD of type E.8 in Proposition 7.3,
・ the 2 fields ( ) with discriminants
have the unique 3-class tower group and 3-tower length ,
・ the single field ( ) with discriminant
has the unique 3-class tower group and 3-tower length .
Among the 11 real quadratic fields with IPOD of type E.9 in Proposition 7.3,
・ the 7 fields ( ) with discriminants
have 3-class tower group or and 3-tower length ,
・ the 4 fields ( ) with discriminants
have 3-class tower group or and 3-tower length .
Proof. Since all these real quadratic fields have 3-capitulation
type or and IPAD , and the 5
fields with have 2nd IPAD
whereas the 9 fields with
have IPAD
the claim is a consequence of Theorem 7.9. □
Remark 7.6. The 3-principalization type E.9 of the field with could not be computed with MAGMA versions up to V2.21-7. Finally, we succeeded to figure it out by means of V2.21-8 [17] .
Figure 5 visualizes sporadic 3-groups of section §7.9 which arise as second 3-class groups of real quadratic fields , , with 3-principalization types D.10, D.5, G.19 and H.4 and the corresponding minimal discriminants, resp. absolute frequencies, which are given in section §7.10 and §7.12.
7.9. Sporadic 3-Groups G of Coclass
Table 5 shows the designation of the transfer kernel type, the IPOD , and the iterated multi-layered IPAD of second order,
for sporadic 3-groups G of type H.4 up to order , characterized by the logarithmic order, , and the SmallGroup identifier, [25] [26] . To enable a brief reference for relative identifiers we put , since this group was called the non-CF group by Ascione [38] [39] .
The groups in Table 5 are represented by vertices of the tree diagram in Figure 6. Figure 6 visualizes sporadic 3-groups of section §7.9 which arise as 3-class tower groups of real quadratic fields , , with 3-principalization type H.4 and the corresponding minimal discriminants, resp. absolute frequencies, in Theorem 7.13 and 7.14.
The tree is infinite, according to Bartholdi, Bush [45] and [ [1] , Cor. 6.2, p. 301].
For and , we can only give the conjectural location of G.
7.10. Real Quadratic Fields of Type H.4
Proposition 7.4 (Fields of type H.4 up to [5] [24] )
In the range of fundamental discriminants of real quadratic fields , there exist precisely 27 cases with 3-principalization type
H.4, , and IPAD . They share the common
second 3-class group .
Proof. The results of [ [24] , Tbl. 6.3, p. 452] were computed in 2010 by means of PARI/GP [14] using an implementation of our principalization algorithm, as described in [ [24] , 5, pp. 446-450]. The frequency 27 in the last column “freq.” for the fourth row concerns type H.4. □
Remark 7.7. To discourage any misinterpretation, we point out that there are four other real quadratic fields with discriminants in the range which possess the same 3-principalization type H.4. However their second 3-class group is isomorphic to either or of order 38, which is not a sporadic group but is located on the coclass tree , and has a different IPAD . The 3-class towers of these fields are determined in [20] .
Theorem 7.13. (3-Class towers of type H.4 up to )
Among the 27 real quadratic fields with type H.4 in Proposition 7.4,
・ the 11 fields ( ) with discriminants
have the unique 3-class tower group and 3-tower length ,
・ the 8 fields ( ) with discriminants
have 3-class tower group or and 3-tower length ,
・ the 5 fields ( ) with discriminants
have the unique 3-class tower group and 3-tower length ,
・ the 3 fields ( ) with discriminants
have a 3-class tower group of order at least 38 and 3-tower length .
Note that , , , and .
Proof. Extensions of absolute degrees 6 and 18 were constructed in steps with MAGMA [17] , using the class field package of C. Fieker [46] . The resulting iterated IPADs of second order were used for the identification, according to Table 5, which is also contained in the more extensive theorem [ [1] , Thm. 6.5, pp. 304-306]. □
7.11. Imaginary Quadratic Fields of Type H.4
Proposition 7.5. (Fields of type H.4 down to [5] [47] )
In the range of fundamental discriminants of imaginary quadratic fields , there exist precisely 6 cases with 3-princi- palization type H.4, , and IPAD . They share the common second 3-class group .
Proof. In the table of suitable base fields [ [47] , p. 84], the row Nr. 4 contains 7 discriminants of imaginary quadratic fields with type H.4. It was computed in 1989 by means of our implementation of the principalization algorithm by Scholz and Taussky, described in [ [47] , pp. 80-83]. In 1989 already, we recognized that only for the discriminant one of the four absolute cubic subfields , , of the unramified cyclic cubic extensions of has 3-class number , which is not the case for the other 6 cases of type H.4 in the table [ [47] , pp. 78-79]. According to [ [5] , Prop. 4.4, p. 485] or [ [5] , Thm. 4.2, p. 489] or [48] , the exceptional cubic field is contained in a sextic field with 3-class number ,
which discourages an IPAD . □
Remark 7.8. The imaginary quadratic field with discriminant possesses the same 3-principalization type H.4, but its second 3-class group is isomorphic to either or of order 38, and has the different IPAD . Results for this field will be given in [20] .
Theorem 7.14. (3-Class towers of type H.4 down to )
Among the 6 imaginary quadratic fields with type H.4 in Proposition 7.5,
・ the 3 fields ( ) with discriminants
have the unique 3-class tower group and 3-tower length ,
・ the 3 fields ( ) with discriminants
have a 3-class tower group of order at least 311 and 3-tower length .
Proof. Using the technique of Fieker [46] , extensions of absolute degrees 6 and 54 were constructed in two steps, squeezing MAGMA [17] close to its limits. The resulting multi-layered iterated IPADs of second order were used for the identification, according to Table 5, resp. the more detailed theorem [ [1] , Thm. 6.5, pp. 304-306]. □
Table 6 shows the designation of the transfer kernel type, the IPOD , and the iterated multi-layered IPAD of second order,
for sporadic 3-groups of type G.19 up to order , characterized by the logarithmic order, , and the SmallGroup identifier, [25] [26] , resp. the relative identifier for . To enable a brief reference for relative id- entifiers we put
, since this group was called the non-CF group by Ascione [39] [38] ,
, , and further
, , and
, , .
The groups in Table 6 are represented by vertices of the tree diagram in Figure 7. Figure 7 visualizes sporadic 3-groups of Table 6 which arise as 3-class tower groups of real quadratic fields , , with 3- principalization type G.19 and the corresponding minimal discriminants, resp. absolute frequencies in Theorem 7.16 and 7.17.
The subtrees are finite and drawn completely for , but they are omitted in the complicated cases , where they reach beyond order .
For , and , we can only give the con- jectural location of .
7.12. Real Quadratic Fields of Type G.19
Proposition 7.6 (Fields of type G.19 up to [5] [24] )
In the range of fundamental discriminants of real quadratic fields , there exist precisely cases with 3-principalization type G.19, , consisting of two disjoint 2-cycles. Their IPAD is uniformly given by , in this range.
Proof. The results of [ [24] , Tbl. 6.3, p. 452] were computed in 2010 by means of PARI/GP [14] using an implementation of our principalization algorithm, as described in [ [24] , 5, pp. 446-450]. The frequency 11 in the last column freq.of the first row concerns type G.19. □
Theorem 7.15. (3-Class towers of type G.19 up to )
The 11 real quadratic fields in Proposition 7.6 with dis- criminants
have the unique 3-class tower group and 3-tower length .
Proof. Extensions of absolute degrees 6 and 18 were constructed with MAGMA [17] , using Fieker’s class field package [46] . The resulting uniform
iterated IPAD of second order was
used for the identification of , according to Table 6. □
Since real quadratic fields of type G.19 seemed to have a very rigid behaviour with respect to their 3-class field tower, admitting no variation at all, we were curious about the continuation of these discriminants beyond the range . Fortunately, M. R. Bush granted access to his extended numerical results for [42] , and so we are able to state the following unexpected answer to our question “Is the 3-class tower group of real quadratic fields with type G.19 and IPAD always isomorphic to in the Small- Groups Library?”
Proposition 7.7. (Fields of type G.19 up to [42] ) In the range of fundamental discriminants of real quadratic fields , there exist precisely 64 cases with 3-principalization type G.19, , and with IPAD .
Proof. Private communication by M. R. Bush [42] . □
Theorem 7.16. (3-Class towers of type G.19 up to )
Among the 64 real quadratic fields with type G.19 in Proposition 7.7,
・ the 11 fields with discriminants in Theorem 7.15 and the 44 fields with discriminants
(that is, together 55 fields or 86%) have ,
the unique 3-class tower group , and 3-tower length ,
・ the 3 fields ( ) with discriminants
have IPAD of second order , the
unique 3-tower group of order , and 3-tower length ,
・ the 6 fields ( ) with discriminants
have iterated IPAD of second order , a 3-class tower
group of order at least , and 3-tower length .
Proof. Similar to the proof of Theorem 7.15, using Table 6, but now applied to the more extensive range of discriminants and various iterated IPADs of second order. □
7.13. Imaginary Quadratic Fields of Type G.19
Proposition 7.8. (Fields of type G.19 down to [5] [24] )
In the range of fundamental discriminants of imaginary quadratic fields , there exist precisely 46 cases with 3-princi- palization type G.19, , consisting of two disjoint 2-cycles, and with IPAD .
Proof. The results of [ [24] , Tbl. 6.4, p. 452] were computed in 2010 by means of PARI/GP [14] using an implementation of our principalization algorithm, as described in [ [24] , 5, pp. 446-450]. The frequency 94 in the last column freq.of the first row concerns type G.19 in the bigger range . Reduced to the first half of this range, we have 46 occurrences. □
Theorem 7.17. (3-Class towers of type G.19 down to )
Among the 46 imaginary quadratic fields with type G.19 in Proposition 7.8,
・ the 30 fields (65%) with discriminants
have iterated IPAD of second order . Conjecturally,
most of them have 3-class tower group of order , and 3-tower length , but and cannot be excluded.
・ The 7 fields (15%) with discriminants
have iterated IPAD of second order , a 3-class
tower group of order at least , and 3-tower length ,
・ the 7 fields (15%) with discriminants
have iterated IPAD of second order , a proven
3-tower group of order , and 3-tower length ,
・ the unique field with discriminant has iterated IPAD of second
order , unknown 3-tower group
and 3-tower length ,
・ the unique field with discriminant has iterated IPAD of
second order , but unknown 3-tower group and
3-tower length .
Proof. Similar to the proof of Theorem 7.15, using Table 6, but now applied to the different range of discriminants and various iterated IPADs of second order.
□
8. Imaginary Quadratic Fields of Type (3,3,3) and Multi-Layered IPADs
In the final section §7 of [1] , we proved that the second 3-class groups of the 14 imaginary quadratic fields with fundamental discriminants and 3-class group of type (3,3,3) are pairwise non-isomorphic [ [1] , Thm. 7.1, p. 307]. For the proof of this theorem in [ [1] , §7.3, p. 311], the IPADs of the 14 fields were not sufficient, since the three fields with discriminants
share the common accumulated (unordered) IPAD
To complete the proof we had to use information on the occupation numbers of the accumulated (unordered) IPODs,
with maximal occupation number 6 for ,
with maximal occupation number 2 for ,
with maximal occupation number 3 for .
Meanwhile we succeeded in computing the second layer of the transfer target type, , for the three critical fields with the aid of the computational algebra system MAGMA [17] by determining the structure of the 3-class groups of the 13 unramified bicyclic bicubic extensions with relative degree and absolute degree 18. In accumulated (unordered) form the second layer of the TTT is given by
for
,
for , and
for .
These results admit incredibly powerful conclusions, which bring us closer to the ultimate goal to determine the precise isomorphism type of . Firstly, they clearly show that the second 3-class groups of the three critical fields are pairwise non-isomorphic without using the IPODs. Secondly, the component with the biggest order establishes an impressively sharpened estimate for the order of from below. The background is explained by the following lemma.
Lemma 8.1. Let be a finite p-group with abelianization of type and denote by the logarithmic order of with respect to the prime number p. Then the abelianizations of subgroups in various layers of G admit lower bounds for :
1) .
2) .
3) , and in particular we have an equation if is metabelian.
Proof. The Lagrange formula for the order of in terms of the index of a subgroup reads
and taking the p-logarithm yields
In particular, we have for , , and again by the Lagrange formula
respectively
with equality if and only if , that is, is abelian.
Finally, is metabelian if and only if is abelian. □
Let us first draw weak conclusions from the first layer of the TTT, i.e. the IPAD, with the aid of Lemma 8.1.
Theorem 8.1. (Coarse estimate [1] )
The order of for the three critical fields K is bounded from below by . If the maximal subgroup with the biggest order of is abelian, i.e. , then the precise logarithmic order of is given by .
Proof. The three critical fields with discriminants
share the common accumulated IPAD
.
Consequently, Lemma 8.1 yields a uniform lower bound for each of the three fields:
The assumption that a maximal subgroup having not the biggest order of were abelian (with ) immediately yields the con- tradiction that
□
It is illuminating that much stronger estimates and conclusions are possible by applying Lemma 8.1 to the second layer of the TTT.
Theorem 8.2. (Finer estimates)
None of the maximal subgroups of for the three critical fields K can be abelian.
The logarithmic order of is bounded from below by
for ,
for ,
for .
Proof. As mentioned earlier already, computations with MAGMA [17] have shown that the accumulated second layer of the TTT is given by
for
,
for , and
for .
Consequently the maximal logarithmic order is
for ,
for ,
for .
According to Lemma 8.1, we have .
Finally, if one of the maximal subgroups of were abelian, then Theorem 8.1 would give the contradiction that . □
Unfortunately, it was impossible for any of the three critical fields K to compute the third layer of the TTT, , that is the structure of the 3-class group of the Hilbert -class field of K, which is of absolute degree 54. This would have given the precise order of the metabelian group , according to Lemma 8.1, since .
We also investigated whether the complete iterated IPAD of second order, , is able to improve the lower bounds in Theorem 8.2 further. It turned out that, firstly none of the additional non-normal components of seems to have bigger order than the normal components of , and secondly, due to the huge 3-ranks of the involved groups, the number of required class group computations enters astronomic regions.
To give an impression, we show the results for five of the 13 maximal subgroups in the case of :
, with 40 components,
, with 121
components,
, with
40 comp.,
,
40 comp.,
, with 40 components.
Acknowledgements
We gratefully acknowledge that our research is supported by the Austrian Science Fund (FWF): P 26008-N25.
Sincere thanks are given to Michael R. Bush (Washington and Lee University, Lexington, VA) for making available numerical results on IPADs of real quadratic fields , and the distribution of discriminants over these IPADs [42] .
We are indebted to Nigel Boston, Michael R. Bush and Farshid Hajir for kindly making available an unpublished database containing numerical results of their paper [13] and a related paper on real quadratic fields, which is still in preparation.
A succinct version of the present article has been delivered on July 09, 2015, within the frame of the 29ièmes Journées Arithmétiques at the University of Debrecen, Hungary [2] .
Funding
Research supported by the Austrian Science Fund (FWF): P 26008-N25.
Cite this paper
Mayer, D.C. (2017) Criteria for Three-Stage Towers of p-Class Fields. Advances in Pure Mathematics, 7, 135-179. https://doi.org/10.4236/apm.2017.72008
References
- 1. Mayer, D.C. (2015) Index-p Abelianization Data of p-Class Tower Groups. Advances in Pure Mathematics, 5, 286-313.
- 2. Mayer, D.C. (2015) Index-p Abelianization Data of p-Class Tower Groups. 29ièmes Journées Arithmétiques (JA 2015), Univ. of Debrecen, Hungary, Presentation Delivered on 9 July 2015.
- 3. Mayer, D.C. (2015) Periodic Sequences of p-Class Tower Groups. Journal of Applied Mathematics and Physics, 3, 746-756.
- 4. Mayer, D.C. (2016) Artin Transfer Patterns on Descendant Trees of Finite p-Groups. Advances in Pure Mathematics, 6, 66-104.
- 5. Mayer, D.C. (2012) The Second p-Class Group of a Number Field. International Journal of Number Theory, 8, 471-505.
- 6. A. Scholz und O. Taussky (1934) Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper: Ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm. Journal für die Reine und Angewandte Mathematik, 171, 19-41.
- 7. Bush, M.R. and Mayer, D.C. (2015) 3-Class Field Towers of Exact Length 3. Journal of Number Theory, 147, 766-777. https://doi.org/10.1016/j.jnt.2014.08.010
- 8. H. Koch und B. B. Venkov, (1975) über den p-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers. Astérisque, 24-25, 57-67.
- 9. Artin, E. (1927) Beweis des allgemeinen Reziprozitätsgesetzes. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 5, 353-363. https://doi.org/10.1007/BF02952531
- 10. Mayer, D.C. (2014) Quadratic p-Ring Spaces for Counting Dihedral Fields. International Journal of Number Theory, 10, 2205-2242.
- 11. Artin, E. (1929) Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 7, 46-51. https://doi.org/10.1007/BF02941159
- 12. Furtwängler, Ph. (1929) Beweis des Hauptidealsatzes für die Klassenkörper algebraischer Zahlkörper. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 7, 14-36. https://doi.org/10.1007/BF02941157
- 13. Boston, N., Bush, M.R. and Hajir, F. (2014) Heuristics for p-Class Towers of Imaginary Quadratic fields. To Appear in Math. Annalen, 2016. (arXiv: 1111.4679v2 [math.NT] 10 Dec 2014.)
- 14. The PARI Group, PARI/GP, Version 2.9.0, Bordeaux, 2016. http://pari.math.u-bordeaux.fr
- 15. Bosma, W., Cannon, J. and Playoust, C. (1997) The Magma Algebra System. I. The User Language. Journal of Symbolic Computation, 24, 235-265. https://doi.org/10.1006/jsco.1996.0125
- 16. Bosma, W., Cannon, J.J., Fieker, C. and Steels, A., Eds. (2016) Handbook of Magma Functions. Edition 2.22, Sydney.
- 17. The MAGMA Group (2016) MAGMA Computational Algebra System. Version 2.22-6, Sydney. http://magma.maths.usyd.edu.au
- 18. Mayer, D.C. (2012) Transfers of Metabelian p-Groups. Monatshefte für Mathematik, 166, 467-495.
- 19. Taussky, O. (1932) über eine Verschärfung des Haupidealsatzes für algebraische Zahlkörper. Journal für die Reine und Angewandte Mathematik, 168, 193-210.
- 20. Mayer, D.C. and Newman, M.F. Finite 3-Groups with Transfer Kernel Type F. In Preparation.
- 21. Mayer, D.C. (2016) p-Capitulation over Number Fields with p-Class Rank Two. Journal of Applied Mathematics and Physics, 4, 1280-1293.
- 22. Mayer, D.C. (2015) New Number Fields with Known p-Class Tower. 22nd Czech and Slovak International Conference on Number Theory (CSICNT 2015), Liptovsky Ján, Slovakia, Presentation Delivered on 31 August 2015.
- 23. Mayer, D.C. (2015) New Number Fields with Known p-Class Tower. Tatra Mountains Mathematical Publications, 64, 21-57.
- 24. Mayer, D.C. (2014) Principalization Algorithm via Class Group Structure. Journal de Théorie des Nombres de Bordeaux, 26, 415-464.
- 25. Besche, H.U., Eick, B. and O’Brien, E.A. (2002) A Millennium Project: Constructing Small Groups. International Journal of Algebra and Computation, 12, 623-644. https://doi.org/10.1142/S0218196702001115
- 26. Besche, H.U., Eick, B. and O’Brien, E.A. (2005) The SmallGroups Library—A Library of Groups of Small Order. An Accepted and Refereed GAP Package, Available Also in MAGMA.
- 27. Newman, M.F. Determination of Groups of Prime-Power Order. In: Group Theory, Canberra, 1975, Lecture Notes in Math., Vol. 573, Springer, Berlin, 1977, 73-84. https://doi.org/10.1007/bfb0087814
- 28. O’Brien, E.A. (1990) The p-Group Generation Algorithm. Journal of Symbolic Computation, 9, 677-698. https://doi.org/10.1016/S0747-7171(08)80082-X
- 29. Mayer, D.C. (2013) The Distribution of Second p-Class Groups on Coclass Graphs. Journal de Théorie des Nombres de Bordeaux, 25, 401-456.
- 30. Mayer, D.C. (2015) Periodic Bifurcations in Descendant Trees of Finite p-Groups. Advances in Pure Mathematics, 5, 162-195.
- 31. Gamble, G., Nickel, W. and O’Brien, E.A. (2006) ANU p-Quotient—p-Quotient and p-Group Generation Algorithms. An Accepted GAP Package, Available Also in MAGMA.
- 32. The GAP Group (2016) GAP—Groups, Algorithms, and Programming—A System for Computational Discrete Algebra. Version 4.8.6, Aachen, Braunschweig, Fort Collins, St. Andrews. http://www.gap-system.org
- 33. Mayer, D.C. (2011) The Distribution of Second p-Class Groups on Coclass Graphs. 27ìemes Journées Arithmétiques (JA 2011), Faculty of Mathematics and Informatics, Univ. of Vilnius, Lithuania, Presentation Delivered on 1 July 2011.
- 34. Mayer, D.C. (2016) Three-Stage Towers of 5-Class Fields. arXiv: 1604.06930v1 [math.NT] 23 Apr 2016.
- 35. Mayer, D.C. (2016) Recent Progress in Determining p-Class Field Towers. Gulf J. Math. (Dubai, UAE). arXiv: 1605.09617v1 [math.NT] 31 May 2016.
- 36. Mayer, D.C. (2016) Recent Progress in Determining p-Class Field Towers. 1st International Colloquium of Algebra, Number Theory, Cryptography and Information Security (ANCI 2016), Taza, Morocco, Invited Keynote Delivered on 12 November 2016.
- 37. Nebelung, B. (1989) Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem. Inauguraldissertation, Universität zu Köln.
- 38. Ascione, J.A., Havas, G. and Leedham-Green, C.R. (1977) A Computer Aided Classification of Certain Groups of Prime Power Order. Bulletin of the Australian Mathematical Society, 17, 257-274, Corrigendum 317-319, Microfiche Supplement, 320.
- 39. Ascione, J.A. (1979) On 3-Groups of Second Maximal Class. Ph.D. Thesis, Australian National University, Canberra.
- 40. F.-P. Heider und B. Schmithals, (1982) Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen. Journal für die Reine und Angewandte Mathematik, 336, 1-25.
- 41. Mayer, D.C. (1991) List of Discriminants of Totally Real Cubic Fields L, Arranged According to Their Multiplicities m and Conductors f. Computer Centre, Department of Computer Science, University of Manitoba, Winnipeg, Canada, Austrian Science Fund, Project Nr. J0497-PHY.
- 42. Bush, M.R. (2015) IPADs of Real Quadratic Fields with 3-Class Rank Two and Discriminants up to 109. 11 July 2015, Private Communication.
- 43. Blackburn, N. (1957) On Prime-Power Groups in Which the Derived Group Has Two Generators. Proceedings of the Cambridge Philosophical Society, 53, 19-27. https://doi.org/10.1017/S0305004100031959
- 44. Shafarevich, I.R. (1964) Extensions with Prescribed Ramification Points (Russian). Publications Mathématiques, Institut des Hautes études Scientiques, 18, 71-95. (English transl. by J. W. S. Cassels in American Mathematical Society Translations, II. Series, 59 (1966), 128-149.)
- 45. Bartholdi, L. and Bush, M.R. (2007) Maximal Unramified 3-Extensions of Imaginary Quadratic Fields and . Journal of Number Theory, 124, 159-166. https://doi.org/10.1016/j.jnt.2006.08.008
- 46. Fieker, C. (2001) Computing Class Fields via the Artin Map. Mathematics of Computation, 70, 1293-1303.
- 47. Mayer, D.C. (1991) Principalization in Complex S3-Fields. Congressus Numerantium 80, 73-87. (Proceedings of the Twentieth Manitoba Conference on Numerical Mathematics and Computing, Univ. of Manitoba, Winnipeg, Canada, 1990.)
- 48. Scholz, A. (1933) Idealklassen und Einheiten in kubischen Körpern. Monatshefte für Mathematik und Physik, 40, 211-222.
Appendix: Corrigenda in [5] [24] [29]
1) The restriction of the numerical results in Proposition 7.1 to the range is in perfect accordance with our machine calculations by means of PARI/GP [14] in 2010, and thus provides the first independent verification of data in [5] [24] [29] .
However, in the manual evaluation of this extensive data material for the ground state of the types a.1, a.2, a.3, and a.3*, a few errors crept in, which must be corrected at three locations: in the tables [ [5] , Tbl. 2, p. 496] and [ [24] , Tbl. 6.1, p. 451], and in the tree diagram [ [29] , Fig. 3.2, p. 422].
The absolute frequency of the ground state is actually given by
1382 instead of the incorrect 1386 for the union of types a.2 and a.3,
698 instead of the incorrect 697 for type a.3*,
2080 instead of the incorrect 2083 for the union of types a.2, a.3, and a.3*, and
150 instead of the incorrect 147 for type a.1.
(The three discriminants were erroneously classified as type a.2 or a.3 instead of a.1.)
In the second table, two relative frequencies (percentages) should be updated:
instead of and
instead of .
2) Incidentally, although it does not concern the section a of IPODs, the single field with discriminant was erroneously classified as type c.18, , instead of H.4, . This has consequences at four locations: in the tables [ [5] , Tbl. 4-5, pp. 498-499] and [ [24] , Tbl. 6.5, p. 452], and in the tree diagram [ [29] , Fig. 3.6, p. 442].
The absolute frequency of these types is actually given by
28 instead of the incorrect 29 for type c.18 (see also [23] , Prop. 7.2]),
4 instead of the incorrect 3 for type H.4.
In the first two tables, the total frequencies should be updated, corres- pondingly:
207 instead of the incorrect 206 in [ [5] , Tbl. 4, p. 498],
66 instead of the incorrect 67 in [ [5] , Tbl. 5, p. 499].