# Pure Cubic Fields 2

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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.
1. Radicand D = 3, conductor f = 32 (kind Ia), u = 1, DPF type γ
2. Radicand D = r with a prime r ≡ 8 (mod 9), conductor f = r (kind II), u = 1, DPF type γ
3. 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
4. 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
5. 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.
1. 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}.
2. 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}.
3. 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}.