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Pure Quintic Fields 2
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§ 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+.
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International Research Project
Towers of p-Class Fields
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