Tables of pure quintic fields

By making use of our generalization of Barrucand and Cohn's theory of principal factorizations in pure cubic fields $\mathbb{Q}(\sqrt[3]{D})$ and their Galois closures $\mathbb{Q}(\zeta_3,\sqrt[3]{D})$ with 3 possible types to pure quintic fields $L=\mathbb{Q}(\sqrt[5]{D})$ and their pure metacyclic normal fields $N=\mathbb{Q}(\zeta_5,\sqrt[5]{D})$ with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields $N$ having normalized radicands in the range $2\le D<10^3$. Our classification is based on the Galois cohomology of the unit group $U_N$, viewed as a module over the automorphism group $\mathrm{Gal}(N/K)$ of $N$ over the cyclotomic field $K=\mathbb{Q}(\zeta_5)$, by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index $(U_K:N_{N/K}(U_N))$ by the number $\#(\mathcal{P}_{N/K} / \mathcal{P}_K)$ of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different $\mathfrak{D}_{N/K}$. The precise structure of the $\mathbb{F}_5$-vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units $(U_N:U_0)$. The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.


Introduction
At the end of his 1975 article on class numbers of pure quintic fields, Parry suggested verbatim "In conclusion the author would like to say that he believes a numerical study of pure quintic fields would be most interesting" [18, p. 484]. Of course, it would have been rather difficult to realize Parry's desire in 1975. But now, 40 years later, we are in the position to use the powerful computer algebra systems PARI/GP [17] and MAGMA [5,6,12] for starting an attack against this hard problem. Prepared by [15,16], this will actually be done in the present paper.
Even in 1991, when we generalized Barrucand and Cohn's theory [4] of principal factorization types from pure cubic fields Q( 3 √ D) to pure quintic fields L = Q( 5 √ D) and their pure metacyclic normal closures N = Q(ζ 5 , 5 √ D) [13], it was still impossible to verify our hypothesis about the distinction between absolute, intermediate and relative differential principal factors (DPF) [15, (6.3)] and about the values of the unit norm index (U K : N N/K (U N )) [15, (1.3)] by actual computations.
All these conjectures have been proven by our most recent numerical investigations. Our classification is based on the Hasse-Iwasawa theorem about the Herbrand quotient of the unit group U N of the Galois closure N of L as a module over the relative group G = Gal(N/K) with respect to the cyclotomic subfield K = Q(ζ 5 ). It only involves the unit norm index (U K : N N/K (U N )) and our 13 types of differential principal factorizations [15,Thm. 1.3], but not the index of subfield units (U N : U 0 ) [15, § 5] in Parry's class number formula [15, (5.1)].
We begin with a collection of explicit multiplicity formulas in § 2 which are required for understanding the subsequent extensive presentation of our computational results in twenty tables of crucial invariants in § 3. This information admits the classification of all 900 pure quintic fields L = Q( 5 √ D) with normalized radicands 2 ≤ D < 10 3 into 13 DPF types and the refined classification into similarity classes with representative prototypes in § 4.
In these final conclusions, we collect theoretical consequences of our experimental results and draw the attention to remaining open questions.

Collection of Multiplicity Formulas
For the convenience of the reader, we provide a summary of formulas for calculating invariants of pure quintic fields L = Q( 5 √ D) with normalized fifth power free radicands D > 1 and their associated pure metacyclic normal fields N = Q(ζ, 5 √ D) with a primitive fifth root of unity ζ = ζ 5 . Let f be the class field theoretic conductor of the relatively quintic Kummer extension N/K over the cyclotomic field K = Q(ζ). It is also called the conductor of the pure quintic field L. The multiplicity m = m(f ) of the conductor f indicates the number of non-isomorphic pure metacyclic fields N sharing the common conductor f , or also, according to [15,Prop. 2.1], the number of normalized fifth power free radicands D > 1 whose fifth roots generate non-isomorphic pure quintic fields L sharing the common conductor f .
The possible DPF types are listed in dependence on U, A, I, R in Table 4, where the symbol × in the column η, resp. ζ, indicates the existence of a unit H ∈ U N , resp. Z ∈ U N , such that η = N N Table 4. Differential principal factorization types, T, of pure metacyclic fields N T U η ζ A I R α 1 Justification of the computational techniques. The steps of the following classification algorithm are ordered by increasing requirements of CPU time. To avoid unnecessary time consumption, the algorithm stops at early stages already, as soon as the DPF type is determined unambiguously. The illustrating subfield lattice of N is drawn in Figure 1 at the end of the paper. Algorithm 3.1. (Classification into 13 DPF types.) Input: a normalized fifth power free radicand D ≥ 2.
Step 1: By purely rational methods, without any number field constructions, the prime factorization of the radicand D (including the counters t, u, v; n, s 2 , s 4 , § 4.2) is determined. If D = q ∈ P, q ≡ ±2 (mod 5), q ≡ ±7 (mod 25), then N is a Polya field of type ε; stop. If D = q ∈ P, q = 5 or q ≡ ±7 (mod 25), then N is a Polya field of type ϑ; stop.
Step 2: The field L of degree 5 is constructed. The primes q 1 , . . . , q T dividing the conductor f of N/K are determined, and their overlying prime ideals q 1 , . . . , q T in L are computed. By means of at most 5 T principal ideal tests of the elements of I L/Q /I Q = T i=1 F 5 q i , the number 5 A := #{(v 1 , . . . , v T ) ∈ F T 5 | T i=1 q vi i ∈ P L }, that is the cardinality of P L/Q /P Q , is determined. If A = T , then N is a Polya field. If A = 3, then N is of type γ; stop. If A = 2, s 2 = s 4 = 0, v ≥ 1, then N is of type ε; stop. If A = 1, s 2 = s 4 = 0, then N is of type ϑ; stop.
Step 5: If the type of the field N is not yet determined uniquely, then U = 1 and there remain the following possibilities. If v ≥ 1, then N is of type δ 1 , if R = 1, of type δ 2 , if I = 1, and of type ε, if R = I = 0. If v = 0, then a fundamental system (E j ) 1≤j≤9 of units is constructed for the unit group U N of the field N of degree 20, and all relative norms of these units with respect to the cyclotomic subfield K are computed. If N N/K (E j ) = ζ k 5 for some 1 ≤ j ≤ 9, 1 ≤ k ≤ 4, then N is of type ζ 1 , if R = 1, of type ζ 2 , if I = 1, and of type η, if R = I = 0. Otherwise the conclusions are the same as for v ≥ 1. Output: the DPF type of the field N = Q(ζ 5 , 5 √ D) and the decision about its Polya property.
Proof. The claims of Step 1 concerning the types ε, ϑ are proved in items (1) and (2)  Step 4, the formulas (4.9) and (4.10) in [15,Thm. 4.4] give an F 5 -basis of the space of relative differential factors, and the formulas (4.11) and (4.12) in [15,Cor. 4.3] determine bounds for the F 5 -dimension R of the space of relative DPF in the field N of degree 20. The claims concerning the types α 1 , α 2 , β 1 are consequences of [15,Thm. 6.1]. Concerning Step 5, the signature of N is (r 1 , r 2 ) = (0, 10), whence the torsion free Dirichlet unit rank of N is given by r = r 1 + r 2 − 1 = 9. The claims about all types are consequences of [15,Thm. 6.1], including information on the constitution of the norm group N N/K (U N ).
Remark 3.1. Whereas the execution of Step 1 and 2 in our Algorithm 3.1, implemented as a Magma program script [12], is a matter of a few seconds on a machine with clock frequency at least 2 GHz, the CPU time for Step 3 lies in the range of several minutes. The time requirement for Step 4 and 5 can reach hours or even days in spite of code optimizations for the calculation of units, in particular the use of the Magma procedures IndependentUnits() and SetOrderUnitsAreFundamental() prior to the call of UnitGroup().
3.4. Conventions and notation in the tables. The normalized radicand D = q e1 1 · · · q es s of a pure metacyclic field N of degree 20 is minimal among the powers D n , 1 ≤ n ≤ 4, with corresponding exponents e j reduced modulo 5. The normalization of the radicands D provides a warranty that all fields are pairwise non-isomorphic [15,Prop. 2.1].
Prime factors are given for composite radicands D only. Dedekind's species, S, of radicands is refined by distinguishing 5 | D (species 1a) and gcd(5, D) = 1 (species 1b) among radicands D ≡ ±1, ±7 (mod 25) (species 1). By the species and factorization of D, the shape of the conductor f is determined. We give the fourth power f 4 to avoid fractional exponents. Additionally, the multiplicity m indicates the number of non-isomorphic fields sharing a common conductor f ( §2). The symbol V F briefly denotes the 5-valuation of the order h F = #Cl(F ) of the class group Cl(F ) of a number field F . By E we denote the exponent of the power in the index of subfield units An asterisk denotes the smallest radicand with given Dedekind kind, DPF type and 5-class groups Cl 5 (F ), F ∈ {L, M, N }. The latter are usually elementary abelian, except for the cases indicated by an additional asterisk (see § 4.4).
Principal factors, P, are listed when their constitution is not a consequence of the other information. According to [15,Thm. 7.2.,item (1)] it suffices to give the rational integer norm of absolute principal factors. For intermediate principal factors, we use the symbols K := L 1−τ = αO M with α ∈ M or L = λO M with a prime element λ ∈ M (which implies L τ = λ τ O M and thus also K = λ 1−τ O M ). Here, (L 1+τ ) 5 = ℓO M when a prime ℓ ≡ ±1 (mod 5) divides the radicand D. For relative principal factors, we use the symbols K 1 := L 1+4τ 2 +2τ +3τ 3 = A 1 O N and Here, (L 1+τ +τ 2 +τ 3 ) 5 = ℓO N when a prime number ℓ ≡ +1 (mod 5) divides the radicand D. (Kernel ideals in [15, § 7].) The quartet (1,2,4,5) indicates conditions which either enforce a reduction of possible DPF types or enable certain DPF types. The lack of a prime divisor ℓ ≡ ±1 (mod 5) together with the existence of a prime divisor q ≡ ±7 (mod 25) and q = 5 of D is indicated by a symbol × for the component 1. In these cases, only the two DPF types γ and ε can occur [15,Thm. 8.1].
A symbol × for the component 2 emphasizes a prime divisor ℓ ≡ −1 (mod 5) of D and the possibility of intermediate principal factors in M , like L and K. A symbol × for the component 4 emphasizes a prime divisor ℓ ≡ +1 (mod 5) of D and the possibility of relative principal factors in N , like K 1 and K 2 . The × symbol is replaced by ⊗ if the facility is used completely, and by (×) if the facility is only used partially.
It is striking that type α 1 with 2-dimensional relative principal factorization, R = 2, and type Similarity classes and prototypes. In [16], we came to the conviction that for deeper insight into the arithmetical structure of the fields under investigation, the prime factorization of the class field theoretic conductor f of the abelian extension N/K over the cyclotomic field K = Q(ζ) and the primary invariants of all involved 5-class groups must be taken in consideration. These ideas have lead to the concept of similarity classes and representative prototypes, which refines the differential principal factorization (DPF) types Let t be the number of primes q 1 , . . . , q t ∈ P distinct from 5 which divide the conductor f . Among these prime numbers, we separately count u := #{1 ≤ i ≤ t | q i ≡ ±1, ±7 (mod 25)} free primes, v := t − u restrictive primes, s 2 := #{1 ≤ i ≤ t | q i ≡ −1 (mod 5)} 2-split primes, and s 4 := #{1 ≤ i ≤ t | q i ≡ +1 (mod 5)} 4-split primes. The multiplicity m = m(f ) is given in terms of t, u, v, according to § 2, and the dimensions of various spaces of primitive ambiguous ideals over the finite field F 5 are given in terms of t, s 2 , s 4 , according to [15, § 4]. By η = 1 2 (1 + √ 5) we denote the fundamental unit of K + = Q(    1, 3; 4). But u = 1 and n = 1 are due to 7, v = 1 and s 4 = 1 are due to 11, in the former case, whereas v = 1 and n = 1 are due to 2, u = 1 and s 4 = 1 are due to 101, in the latter case. Therefore, the contributions by primes congruent to ±1 (mod 25) will be indicated by writing u = 1 ′ and s 4 = 1 ′ , resp. s 2 = 1 ′ .
We also emphasize that in the rare cases of non-elementary 5-class groups, the actual structures (abelian type invariants) of the 5-class groups will be taken into account, and not only the 5valuations V L , V M , V N . The remaining elements of a similarity class, which are bigger than the prototype, only reproduce the arithmetical invariants of the prototype and do not provide any additional information, exept possibly about other primary components of the class groups, that is the structure of ℓ-class groups Cl ℓ (F ) of the fields F ∈ {L, M, N } for ℓ ∈ P \ {5}.
Whereas there are only 13 DPF types of pure quintic fields, the number of similarity classes is obviously infinite, since firstly the number t of primes dividing the conductor is unbounded and secondly the number of states, defined by the triplet (V L , V M , V N ) of 5-valuations of class numbers, is also unlimited.
Given a fixed refined Dedekind species (e 0 ; t, u, v, m; n, s 2 , s 4 ), the set of all associated normalized fifth power free radicands D usually splits into several similarity classes defined by distinct DPF types (type splitting). Occasionally it even splits further into different structures of 5-class groups, called states, with increasing complexity of abelian type invariants (state splitting).
In fact, the shape of the conductors in Theorem 4.1 does not only determine the refined Dedekind species and the DPF type, but also the structure of the 5-class groups of the fields L, M and N .  The pure metacyclic fields N associated with these four similarity classes are Polya fields. For similarity classes distinct from the four infinite classes in Theorem 4.1 we cannot provide deterministic criteria for the DPF type and for the homogeneity of multiplets with m > 1. In general, the members of a multiplet belong to distinct similarity classes, thus giving rise to heterogeneous DPF types. We explain these phenomena with the simplest cases where only two DPF types are involved (type splitting). (1) f 4 = 5 6 · q 4 with q ∈ P, q ≡ ±7 (mod 25) gives rise to a quartet with possibly heterogeneous DPF type (ε x , η y ), x + y = 4, (2) f = q 1 · q 2 with q i ∈ P, q i ≡ ±7 (mod 25) gives rise to a quartet with possibly heterogeneous DPF type (ε x , η y ), x + y = 4.
Proof. This is a consequence of [15, Thm. 10.5 and Thm. 6.1], taking into account that the prime 5 is not included in the current definition of the counter t (with value t = 1 in the present situation), and thus the estimate in [15,Cor. 4 Inspired by the last two theorems, it is worth ones while to summarize, for each kind of prime radicands, what is known about the possibilities for differential principal factorizations.
Secondly, for a prime radicand D ≡ ±1 (mod 5) which splits in M , the space of radicals ∆ = 5 √ D is a 1-dimensional subspace of absolute DPF contained in the 2-dimensional space ∆ ⊕ ∆ ′ of differential factors generated by the two prime ideals of M over D. Consequently, in this special situation there arises an additional constraint I ≤ 1 for the dimension of the space of intermediate DPF, which must be contained in the 1-dimensional complement ∆ ′ . This generally excludes type α 3 with I = 2 for prime radicands.

Refinement of DPF types by similarity classes.
Based on the definition of similarity classes and prototypes in § 4.2, on the explicit listing of all prototypes in the range between 2 and 10 3 in the Tables 27 -30, and on theoretical foundations in § 4.3, we are now in the position to establish the intended refinement of our 13 differential principal factorization types into similarity classes in the Tables 31 -43, as far as the range of our computations for normalized radicands 2 ≤ D < 10 3 is concerned. The cardinalities |M| refine the statistical evaluation in Table 26. DPF types are characterized by the multiplet (U, η, ζ; A, I, R), refined Dedekind species, S, by the multiplet (e 0 ; t, u, v, m; n, s 2 , s 4 ), and 5-class groups by the multiplet (V L , V M , V N ; E).   (2,3,6). Type α 2 with E = 1 splits into 7 similarity classes in the ground state (V L , V M , V N ) = (2, 2, 4) and 4 similarity classes in the first excited state (V L , V M , V N ) = (3,4,8). Summing up the partial frequencies 40 + 7, resp. 24 + 4, of these states in Table 32 yields the considerable absolute frequency 75 of type α 2 in the range 2 ≤ D < 10 3 , as given in Table 26. Type α 2 is the unique type with mixed intermediate and relative principal factorization, I = R = 1. . The modest absolute frequency 7 of type η in the range 2 ≤ D < 10 3 , given in Table 26, is the sum 6 + 1 of partial frequencies in Table  42. Type η only occurs with logarithmic subfield unit index E = 6. It is a type with 2-dimensional absolute principal factorization, A = 2. However, it should be pointed out that outside of the range of our systematic investigations we found an excited state (V L , V M , V N ) = (1, 2, 5) for the similarity class [1505], where 1505 = 5 · 7 · 43 has three prime divisors, additionally to the ground state (V L , V M , V N ) = (0, 0, 1).  DPF type ϑ splits into the unique finite similarity class [5] with only a single element and the infinite parametrized sequence [7] consisting of all prime radicands D = q congruent to ±7 (mod 25). The small absolute frequency 19 of type ϑ in the range 2 ≤ D < 10 3 , given in Table 26, is the sum |5| + |7| = 1 + 18 in Table 43. Since no theoretical argument disables the occurrence of type ϑ for composite radicands D with prime factors 5 and q ≡ ±7 (mod 25), we conjecture that such cases will appear in bigger ranges with D > 10 3 . Type ϑ only occurs with logarithmic subfield unit index E = 5, and is the unique type where every unit of K occurs as norm of a unit of N , that is U = 0. 4.6. Increasing dominance of DPF type γ for T → ∞. In this final section, we want to show that the careful book keeping of similarity classes with representative prototypes in the Tables 31 -43 is useful for the quantitative illumination of many other phenomena. For an explanation, we select the phenomenon of absolute principal factorizations.
The statistical distribution of DPF types in Table 26 has proved that type γ with 324 occurrences, that is 36 %, among all 900 fields N = Q(ζ 5 , 5 √ D) with normalized radicands in the range 2 ≤ D < 10 3 is doubtlessly the high champion of all DPF types. This means that there is a clear trend towards the maximal possible extent of 3-dimensional spaces of absolute principal factorizations, A = 3, in spite of the disadvantage that the estimate 1 ≤ A ≤ min(3, T ) in the formulas (4.3) and (4.4) of [15,Cor. 4.1] prohibits type γ for conductors f with T ≤ 2 prime divisors.
For the following investigation, we have to recall that the number T of all prime factors of f 4 = 5 e0 · q 4 1 . . . q 4 t is given by T = t + 1 for fields of Dedekind's species 1, where e 0 ∈ {2, 6}, and by T = t for fields of Dedekind's species 2, where e 0 = 0.
Conductors f with T = 4 prime factors occur in six tables, 1 case of type α 2 in a single similarity class of Table 32, 1 case of type α 3 in a single similarity class of Table 33, 7 cases of type β 1 in a single similarity class of Table 34, 10 cases of type β 2 in 5 similarity classes of Table 35, 126 cases of type γ in 16 similarity classes of Table 36, 8 cases of type ε in 3 similarity classes of Table 39, that is, a total of 153 cases, with respect to the complete range 2 ≤ D < 10 3 of our computations. Consequently, we have an increase of type γ from 36.0 %, with respect to the entire database, to 126 153 = 82.4 %, with respect to T = 4.
The feature is even aggravated for conductors f with T = 5 prime factors, which exclusively occur in , with a total of 6 elements, all (100 %) with associated fields of type γ.

Acknowledgements
We gratefully acknowledge that our research was supported by the Austrian Science Fund (FWF): projects J 0497-PHY and P 26008-N25. This work is dedicated to the memory of Charles J. Parry ( † 25 December 2010) who suggested a numerical investigation of pure quintic number fields.