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Pure Cubic Fields
§ 1. Galois closure and subfield lattice.
Let q1,…,qs be pairwise distinct primes
such that s ≥ 1 and 3 may be among them.
Denote by L = Q(D1/3) the pure cubic number field
with third power free radicand D = q1e1 … qses,
where the exponents are integers 1 ≤ ej ≤ 2.
The field L is generated by adjoining the unique real solution
of the pure cubic equation X3 - D = 0 to the rational number field Q.
It is a non-Galois algebraic number field with signature (1,1)
and thus possesses two isomorphic complex fields
Lj = Q(ζjD1/3),
1 ≤ j ≤ 2,
all of whose arithmetical invariants coincide.
The normal closure of L is the compositum N = Q(D1/3,ζ)
of L = Q(D1/3) with the cyclotomic field k = Q(ζ),
where ζ = ζ3 = exp(2 π i / 3) denotes a primitive third root of unity.
N is a complex dihedral field of degree 6 whose Galois group
Gal(N|Q)
is the semidirect product D3 = C3 * C2 of two cyclic groups
and can also be viewed as a symmetric group S3 or a metacyclic group M3.
The cyclotomic field k is a complex quadratic field.
§ 2. Class numbers and unit index.
In 1971, Barrucand and Cohn have determined the class number relation
3 hN = (UN:U0) hL2
[Thm. 14.1, p. 232, BaCo2],
where
UN denotes the unit group of N,
U0 is the subgroup of UN generated by
all units of the conjugate fields
Lj = Q(ζjD1/3),
0 ≤ j ≤ 2,
of L and of k,
and the unit index (UN:U0) can take
two possible values 3e with 0 ≤ e ≤ 1
[Thm. 12.1, p. 229, BaCo2].
At the end of their papers
[§ 8, p. 19, BaCo1] and [§ 17, pp. 237 - 238, BaCo2],
Barrucand and Cohn present a classification
of some pure cubic fields L with small radicands D between 2 and 103.
Examples 2.1.
We give the smallest radicands D of pure cubic fields L = Q(D1/3)
where the two values of the exponent e actually occur:
e = 1 for D = 2 of type β,
e = 0 for D = 7 of type α.
In the last case, UN is generated completely by subfield units.
§ 3. Conductor and discriminants.
Let R = q1 … qs be the squarefree product
of all prime divisors of the radicand D
of the pure cubic field L = Q(D1/3).
Independently of the exponents e1,…,es,
the conductor f of the cyclic cubic relative extension N|k
is given by
f = 3 R
if not D ≡ ±1 (mod 9) (field of Dedekind's first kind),
f = R
if D ≡ ±1 (mod 9) (field of Dedekind's second kind)
[De], [Thm. 1, p. 103, Ma1].
Examples 3.1.
The smallest radicands D of pure cubic fields L = Q(D1/3)
with given kind and increasing number of prime divisors are
D = 2, 14 = 2*7, for fields of Dedekind's first kind with (D,3) = 1,
D = 3, 6 = 2*3, 30 = 2*3*5 , for fields of Dedekind's first kind with 3|D,
D = 17, 10 = 2*5, for fields of Dedekind's second kind.
It is well known that the cyclotomic discriminant takes the value
d(k) = -3,
and Hilbert's Theorem 39 on discriminants of composite fields shows the following
Theorem 3.1.
The dihedral discriminant is given by
d(N) = d(k)3 f4 =
= -37 R4 for a field of Dedekind's first kind,
= -33 R4 for a field of Dedekind's second kind,
and the pure cubic discriminant is given by
d(L) = d(k) f2 =
= -33 R2 for a field of Dedekind's first kind,
= -3 R2 for a field of Dedekind's second kind.
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International Research Project
Towers of p-Class Fields
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