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Pure Cubic Fields 2
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§ 6. DPF types of pure cubic fields.
We define three possible differential principal factorization types
of pure cubic number fields L = Q(D^{1/3}),
according to the unit norm index (U_{k}:Norm_{Nk}U_{N})
as the primary invariant
and the pair (a,r) consisting of
the order a = #P_{a} of the group of absolute DPF of LQ, and
the index r = (P_{r}:P_{a}) in the group of relative DPF of Nk,
as the secondary invariant
(similar to but not identical with our definitions in [Ma2]).
The connection between the various quantities is given by the chain of equations
a * r = #P_{r} = (P_{N}^{Gal(Nk)}:P_{k}) =
(E_{Nk}:U_{N}^{1  σ}) =
3 (U_{k}:Norm_{Nk}U_{N}).
Note that the unit index
(U_{N}:U_{0})
does not enter our definition of the DPF types.
However, Barrucand and Cohn
[Thm. 15.6, pp. 235  236, BaCo2]
define four types of pure cubic fields L,
according to the unit index
(U_{N}:U_{0}),
the pair (a,r),
and the representation of a fundamental unit ε of L with norm Norm_{Nk}ε = +1
as ε = A^{1  σ} with an algebraic integer A of N,
in the sense of Hilbert's Theorem 90.
HalterKoch [HK] has proved that type II is actually impossible.
(U_{k}:Norm_{Nk}U_{N}) 
(P_{N}^{Gal(Nk)}:P_{k}) 
a 
r 
DPF type 
type 
(U_{N}:U_{0}) 
Norm_{Nk}A 
3 
9 
3 
3 
α 
III 
1 
nonunit 
3 
9 
3 
3 
impossible 
II 
3 
1 
3 
9 
9 
1 
β 
I 
3 
1 
1 
3 
3 
1 
γ 
IV 
3 
ζ^{±1} 
§ 7. Computational results.
With the aid of PARI/GP and MAGMA we have determined the DPF type
and other invariants of the 35 pure cubic number fields L = Q(D^{1/3})
with normalized radicands D = q_{1}^{e1} … q_{s}^{es}
(minimal among the powers D^{n}, 1 ≤ n ≤ 2, with corresponding e_{j}s reduced mod 3)
in Barrucand and Cohn's range 2 ≤ D < 50
[§ 17, pp. 237  238, BaCo2].
Prime factors are given for composite D only.
The first kind of fields is refined by distinguishing 3D (kind Ia) and (3,D) = 1 (kind Ib).
The exponent of the power in the unit index (U_{N}:U_{0}) = 3^{e}
is denoted by e.
The unit norm index (U_{k}:Norm_{Nk}U_{N}) is abbreviated by u.
The symbol Cl briefly denotes the 3class group Cl_{3} of a number field.
An asterisk denotes the smallest radicand with given type and 3class groups.
No. 
D 
factors 
kind 
Cl(L) 
Cl(N) 
e 
u 
DPF type 
* 
1 
2 

Ib 
1 
1 
1 
3 
β 
* 
2 
3 

Ia 
1 
1 
1 
1 
γ 
* 
3 
5 

Ib 
1 
1 
1 
3 
β 
4 
6 
2,3 
Ia 
1 
1 
1 
3 
β 
5 
7 

Ib 
(3) 
(3) 
0 
3 
α 
* 
6 
10 
2,5 
II 
1 
1 
1 
3 
β 
7 
11 

Ib 
1 
1 
1 
3 
β 
8 
12 
3,2^{2} 
Ia 
1 
1 
1 
3 
β 
9 
13 

Ib 
(3) 
(3) 
0 
3 
α 
10 
14 
2,7 
Ib 
(3) 
(3,3) 
1 
3 
β 
* 
11 
15 
3,5 
Ia 
1 
1 
1 
3 
β 
12 
17 

II 
1 
1 
1 
1 
γ 
13 
19 

II 
(3) 
(3) 
0 
3 
α 
14 
20 
5,2^{2} 
Ib 
(3) 
(3,3) 
1 
3 
β 
15 
21 
3,7 
Ia 
(3) 
(3) 
0 
3 
α 
16 
22 
2,11 
Ib 
(3) 
(3,3) 
1 
3 
β 
17 
23 

Ib 
1 
1 
1 
3 
β 
18 
26 
2,13 
II 
(3) 
(3) 
0 
3 
α 
19 
28 
7,2^{2} 
II 
(3) 
(3) 
0 
3 
α 
20 
29 

Ib 
1 
1 
1 
3 
β 
21 
30 
2,3,5 
Ia 
(3) 
(3,3) 
1 
3 
β 
22 
31 

Ib 
(3) 
(3) 
0 
3 
α 
23 
33 
3,11 
Ia 
1 
1 
1 
3 
β 
24 
34 
2,17 
Ib 
(3) 
(3,3) 
1 
3 
β 
25 
35 
5,7 
II 
(3) 
(3) 
0 
3 
α 
26 
37 

II 
(3) 
(3) 
0 
3 
α 
27 
38 
2,19 
Ib 
(3) 
(3,3) 
1 
3 
β 
28 
39 
3,13 
Ia 
(3) 
(3) 
0 
3 
α 
29 
41 

Ib 
1 
1 
1 
3 
β 
30 
42 
2,3,7 
Ia 
(3) 
(3,3) 
1 
3 
β 
31 
43 

Ib 
(3) 
(3) 
0 
3 
α 
32 
44 
11,2^{2} 
II 
1 
1 
1 
3 
β 
33 
45 
5,3^{2} 
Ia 
1 
1 
1 
3 
β 
34 
46 
2,23 
II 
1 
1 
1 
3 
β 
35 
47 

Ib 
1 
1 
1 
3 
β 
§ 8. Theoretical results.
The following theorem extends
[Thm., p. 8, Ho]
and
[Thm. 8, p. 223, PaWa1],
resp.
[PaWa2],
by giving the conductor f, the unit norm index
u = (U_{k}:Norm_{Nk}U_{N}),
and the differential principal factorization (DPF) type.
Theorem 8.1.
The 3class group of the Galois closure N of a pure cubic field L = Q(D^{1/3}) is trivial,
Cl_{3}(N) = 1,
if and only if one of the following five conditions is satisfied.

Radicand D = 3, conductor f = 3^{2} (kind Ia), u = 1, DPF type γ

Radicand D = r with a prime r ≡ 8 (mod 9), conductor f = r (kind II), u = 1, DPF type γ

Radicand D = q with a prime q ≡ 2 (mod 3) but not ≡ 8 (mod 9),
conductor f = 3 q (kind Ib), u = 3, DPF type β

Radicand D = q^{e1} 3^{e2} with a prime q ≡ 2 (mod 3) but not ≡ 8 (mod 9),
conductor f = 3^{2} q (kind Ia), u = 3, DPF type β

Radicand D = q^{e1} r^{e2} ≡ ±1 (mod 9)
with primes q,r ≡ 2 (mod 3)
not both ≡ 8 (mod 9), conductor f = q r (kind II), u = 3, DPF type β
The exponent of the power in the unit index (U_{N}:U_{0}) = 3^{e}
takes the value e = 1 in all cases.
Furthermore, Cl_{3}(N) = 1 is equivalent to Cl_{3}(L) = 1.
Examples 8.1.
Using our numerical results in section 7,
we give all radicands D < 50 of pure cubic fields L = Q(D^{1/3})
where the various configurations of Theorem 8.1 actually occur.

A single prime divisor of the radicand D and of the conductor f:
D = r ≡ 8 (mod 9) of second kind, f = r, for
D ∈ {17}.

A single prime divisor of the radicand D and two prime divisors of the conductor f:
D = q ≡ 2,5 (mod 9) of first kind (Ib), f = 3 q, for
D ∈ {2,5,11,23,29,41,47}.

Two prime divisors of the radicand D and of the conductor f:
D = q^{e1} 5^{e2} of first kind (Ia), f = 3^{2} q, for
D ∈ {6,12,15,33,45},
D = q^{e1} r^{e2} of second kind, f = q r, for
D ∈ {10,44,46}.



Trade, Science, Art and Industry
Principal investigator of the
International Research Project
Towers of pClass Fields
over Algebraic Number Fields
supported by the Austrian Science Fund (FWF):
P 26008N25
