# Pure Cubic Fields

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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. Next Page