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Pure Quintic Fields 2
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§ 5. Ambiguous principal ideals.
Since the cyclotomic unit group U_{k} is generated by < 1, ζ, η >,
where η > 1 denotes the fundamental unit of Q(5^{1/2}),
and since
Norm_{Nk}U_{N} ≥ Norm_{Nk}U_{k} = U_{k}^{5}
= < 1, η^{5} >,
we obtain bounds for the unit norm index
(U_{k}:Norm_{Nk}U_{N}) ∈ {1,5,25},
according to whether Norm_{Nk}U_{N} contains {ζ,η}
or only η (resp. ζ) or none of them.
The somewhat abstract quotient E_{Nk}/U_{N}^{1  σ}
is isomorphic to the more ostensive quotient
P_{N}^{Gal(Nk)}/P_{k}
of the group of ambiguous principal ideals of Nk
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
(P_{N}^{Gal(Nk)}:P_{k}) =
(E_{Nk}:U_{N}^{1  σ}) =
5 (U_{k}:Norm_{Nk}U_{N}) ∈ {5,25,125}.
Even in the worst case (P_{N}^{Gal(Nk)}:P_{k}) = 5,
there are at least the radicals D^{1/5}, … ,D^{4/5}, and the unit 1
which generate 5 distinct ambiguous principal ideals of LQ.
Examples 5.1.
Anticipating some of our computational results in section 8,
we give the smallest radicands D of pure quintic fields L = Q(D^{1/5})
where the various values of #P_{r} = (P_{N}^{Gal(Nk)}:P_{k}) actually occur:
#P_{r} = 5 for D = 5 of type θ,
#P_{r} = 25 for D = 2 of type ε,
#P_{r} = 125 for D = 6 = 2*3 of type γ.
Generally, the fixed value of the Herbrand quotient
for a given type of field extension Nk
can be interpreted by the following
Tradeoff Principle.
If many units in U_{k} can be represented as norms of units in U_{N},
then the extension Nk 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
P_{N} ∩ I_{k}/P_{k} ≤ P_{N}^{Gal(Nk)}/P_{k},
the socalled capitulation kernel of Nk is trivial, because h(k) = 1.
However, the elementary abelian 5group P_{r} = P_{N}^{Gal(Nk)}/P_{k},
whose generators are principal ideals dividing the different of Nk
(socalled differential principal factors),
consists of three nested subgroups, P_{r} ≥ P_{i} ≥ P_{a}
(similar to but not identical with our definitions in [Ma2]),

absolute DPF of LQ, P_{a}, always containing the above mentioned radicals,

intermediate DPF of Mk^{+}, P_{i} \ P_{a},
such that the proper cosets of P_{a} in P_{i}
do not project down to LQ by taking the norm, and

relative DPF of Nk, P_{r} \ P_{i},
such that the proper cosets of P_{i} in P_{r}
do not even project down to Mk^{+} 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 ∈ Norm_{Nk}U_{N}
can occur only when f is divisible by no other primes than
5 and / or primes q_{j} ≡ ±1,±7 (mod 25).

Relative DPF of Nk can occur only if
some prime q_{j} ≡ +1 (mod 5) divides f.

Intermediate DPF of Mk^{+} can occur only if
some prime q_{j} ≡ ±1 (mod 5) divides f.

Absolute DPF of LQ, 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 ∈ Norm_{Nk}U_{N} is due to
the properties of the quintic Hilbert symbol (fifth power norm residue symbol) over k.
2. The condition for relative DPF of Nk 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 Mk^{+} 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^{+}.
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over Algebraic Number Fields
supported by the Austrian Science Fund (FWF):
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