Fame For Styria 2014: Pure Quintic Fields 2

Pure Quintic Fields 2

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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.
  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).
  2. Relative DPF of N|k can occur only if some prime qj ≡ +1 (mod 5) divides f.
  3. Intermediate DPF of M|k+ can occur only if some prime qj ≡ ±1 (mod 5) divides f.
  4. 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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Trade, Science, Art and Industry
Daniel C. Mayer
Principal investigator of the
International Research Project
Towers of p-Class Fields
over Algebraic Number Fields
supported by the Austrian Science Fund (FWF):
P 26008-N25

Bibliographical References:

[Ma2] D. C. Mayer,
Classification of dihedral fields,
University of Manitoba, Winnipeg, Manitoba, Canada, 1991.

Web master's e-mail address:

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