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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(D1/3),
according to the unit norm index (Uk:NormN|kUN)
as the primary invariant
and the pair (a,r) consisting of
the order a = #Pa of the group of absolute DPF of L|Q, and
the index r = (Pr:Pa) in the group of relative DPF of N|k,
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 = #Pr = (PNGal(N|k):Pk) =
(EN|k:UN1 - σ) =
3 (Uk:NormN|kUN).
Note that the unit index
(UN:U0)
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
(UN:U0),
the pair (a,r),
and the representation of a fundamental unit ε of L with norm NormN|kε = +1
as ε = A1 - σ with an algebraic integer A of N,
in the sense of Hilbert's Theorem 90.
Halter-Koch [HK] has proved that type II is actually impossible.
| (Uk:NormN|kUN) |
(PNGal(N|k):Pk) |
a |
r |
DPF type |
type |
(UN:U0) |
NormN|kA |
| 3 |
9 |
3 |
3 |
α |
III |
1 |
non-unit |
| 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(D1/3)
with normalized radicands D = q1e1 … qses
(minimal among the powers Dn, 1 ≤ n ≤ 2, with corresponding ejs 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 3|D (kind Ia) and (3,D) = 1 (kind Ib).
The exponent of the power in the unit index (UN:U0) = 3e
is denoted by e.
The unit norm index (Uk:NormN|kUN) is abbreviated by u.
The symbol Cl briefly denotes the 3-class group Cl3 of a number field.
An asterisk denotes the smallest radicand with given type and 3-class 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 |
3,22 |
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 |
5,22 |
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 |
7,22 |
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 |
11,22 |
II |
1 |
1 |
1 |
3 |
β |
| 33 |
45 |
5,32 |
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 = (Uk:NormN|kUN),
and the differential principal factorization (DPF) type.
Theorem 8.1.
The 3-class group of the Galois closure N of a pure cubic field L = Q(D1/3) is trivial,
Cl3(N) = 1,
if and only if one of the following five conditions is satisfied.
-
Radicand D = 3, conductor f = 32 (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 = qe1 3e2 with a prime q ≡ 2 (mod 3) but not ≡ 8 (mod 9),
conductor f = 32 q (kind Ia), u = 3, DPF type β
-
Radicand D = qe1 re2 ≡ ±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 (UN:U0) = 3e
takes the value e = 1 in all cases.
Furthermore, Cl3(N) = 1 is equivalent to Cl3(L) = 1.
Examples 8.1.
Using our numerical results in section 7,
we give all radicands D < 50 of pure cubic fields L = Q(D1/3)
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 = qe1 5e2 of first kind (Ia), f = 32 q, for
D ∈ {6,12,15,33,45},
D = qe1 re2 of second kind, f = q r, for
D ∈ {10,44,46}.
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