# Pure Quintic Fields 2

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 Pure Quintic Fields 2 Previous Page § 5. Ambiguous principal ideals. Since the cyclotomic unit group Uk is generated by < -1, ζ, η >, where η > 1 denotes the fundamental unit of Q(51/2), and since NormN|kUN ≥ NormN|kUk = Uk5 = < -1, η5 >, we obtain bounds for the unit norm index (Uk:NormN|kUN) ∈ {1,5,25}, according to whether NormN|kUN contains {ζ,η} or only η (resp. ζ) or none of them. The somewhat abstract quotient EN|k/UN1 - σ is isomorphic to the more ostensive quotient PNGal(N|k)/Pk of the group of ambiguous principal ideals of N|k modulo the subgroup of principal ideals of k, according to Iwasawa (or also to Hilbert's Theorems 92 and 94), and by the Takagi/Hasse Theorem on the Herbrand quotient, we have (PNGal(N|k):Pk) = (EN|k:UN1 - σ) = 5 (Uk:NormN|kUN) ∈ {5,25,125}. Even in the worst case (PNGal(N|k):Pk) = 5, there are at least the radicals D1/5, … ,D4/5, and the unit 1 which generate 5 distinct ambiguous principal ideals of L|Q. Examples 5.1. Anticipating some of our computational results in section 8, we give the smallest radicands D of pure quintic fields L = Q(D1/5) where the various values of #Pr = (PNGal(N|k):Pk) actually occur: #Pr = 5 for D = 5 of type θ, #Pr = 25 for D = 2 of type ε, #Pr = 125 for D = 6 = 2*3 of type γ. Generally, the fixed value of the Herbrand quotient for a given type of field extension N|k can be interpreted by the following Trade-off Principle. If many units in Uk can be represented as norms of units in UN, then the extension N|k contains only few ambiguous principal ideals, and vice versa (by interchanging "many" and " only few"). § 6. Differential principal factorizations (DPF). As opposed to Hilbert's Theorem 94, the subgroup PN ∩ Ik/Pk ≤ PNGal(N|k)/Pk, the so-called capitulation kernel of N|k is trivial, because h(k) = 1. However, the elementary abelian 5-group Pr = PNGal(N|k)/Pk, whose generators are principal ideals dividing the different of N|k (so-called differential principal factors), consists of three nested subgroups, Pr ≥ Pi ≥ Pa (similar to but not identical with our definitions in [Ma2]), absolute DPF of L|Q, Pa, always containing the above mentioned radicals, intermediate DPF of M|k+, Pi \ Pa, such that the proper cosets of Pa in Pi do not project down to L|Q by taking the norm, and relative DPF of N|k, Pr \ Pi, such that the proper cosets of Pi in Pr do not even project down to M|k+ by taking the norm. The existence, resp. the lack, of certain prime divisors of the conductor f permits some criteria for the classification of pure quintic fields. Theorem 6.1. ζ ∈ NormN|kUN can occur only when f is divisible by no other primes than 5 and / or primes qj ≡ ±1,±7 (mod 25). Relative DPF of N|k can occur only if some prime qj ≡ +1 (mod 5) divides f. Intermediate DPF of M|k+ can occur only if some prime qj ≡ ±1 (mod 5) divides f. Absolute DPF of L|Q, distinct from radicals, must exist when no prime ≡ ±1 (mod 5) divides f and f has at least one prime factor distinct from 5 and from primes ≡ ±7 (mod 25). Remarks 6.1. 1. The condition for ζ ∈ NormN|kUN is due to the properties of the quintic Hilbert symbol (fifth power norm residue symbol) over k. 2. The condition for relative DPF of N|k is a consequence of the decomposition law (e,f,g) = (1,1,4) for primes ≡ +1 (mod 5) in k. 3. The condition for intermediate DPF of M|k+ is due to the decomposition types (e,f,g) = (1,1,2), resp. (1,2,2), of primes ≡ -1 (mod 5) in k+, resp. k, and (e,f,g) = (1,1,2) of primes ≡ +1 (mod 5) in k+. Next Page