Our Mission:
to advance Austrian Science
to the Forefront of International Research
and to stabilize this status

Pure Quintic Fields 4
Previous Page
§ 9. Algorithmic techniques.

1^{st} step, the unit index (U_{N}:U_{0}):
With the aid of PARI/GP, we construct the fields L, M and N of degrees 5, 10 and 20,
and we compute the structure of the 5class groups Cl_{5}(L), Cl_{5}(M) and Cl_{5}(N).
Then we use Parry's class number relation to calculate the unit index
(U_{N}:U_{0}) = 5^{5} h_{N} / h_{L}^{4}
where h_{L} = #Cl_{5}(L) and h_{N} = #Cl_{5}(N).
(This step executes rather quickly in about 3 hours of CPU time
for the 90 fields with D ≤ 110 of section 8.)

2^{nd} step, the ambiguous principal ideals:
By means of MAGMA we compute all ideals which divide the relative different of Nk
and perform a principal ideal test for each of them.
(This step executes very rapidly, even for big radicands D > 100.)

3^{rd} step, the unit norm index (U_{k}:Norm_{Nk}U_{N}):
Since the normal closure N of L is of signature (0,10),
its torsion free unit rank is 0+101 = 9,
and we let MAGMA determine a fundamental system of 9 units
and their relative norms w. r. t. k.
As a supplement, we also construct the real intermediate field M of signature (2,4),
and we calculate the relative norms w. r. t. k^{+} of the 2+41 = 5 fundamental units.
It turns out that, if η is relative norm of a unit of N, then also of a unit of M.
(Unfortunately, for radicands D bigger than 25,
this step needs more than a week of CPU time for a single field.)
§ 10. Theoretical foundations.
For enumerating ambiguous ideals and classes we need the following
counters of primes dividing the conductor.
t = #{ q prime distinct from 5  q divides f },
w = 1 if 5 divides f, w = 0 otherwise,
τ = #{ q prime  q divides f } = t + w;
primes which do not split in k:
n = #{ q ≡ ±2 (mod 5) prime  q divides f },
ν = #{ q prime  q divides f } = n + w;
primes which split in k:
s = #{ q ≡ 1 (mod 5) prime  q divides f },
σ = #{ q ≡ +1 (mod 5) prime  q divides f }.
Proposition 10.1.
The number of all prime ideals
1. of Q which are totally ramified in L is τ = ν + s + σ,
2. of k^{+} which are totally ramified in M is δ = ν + 2s + 2σ,
3. of k which are totally ramified in N is d = ν + 2s + 4σ.
Let G = Gal(Nk), then the number of all
4. absolutely ambiguous ideals of LQ is #(I_{L}^{G} / I_{Q}) = 5^{τ},
where
I_{L}^{G}I_{k} / I_{k} ≅ I_{L}^{G} / I_{L}∩I_{k} = I_{L}^{G} / I_{Q},
5. intermediate ambiguous ideals of Mk^{+} is #(I_{M}^{G} / I_{k+}) = 5^{δ},
where
I_{M}^{G}I_{k} / I_{k} ≅ I_{M}^{G} / I_{M}∩I_{k} = I_{L}^{G} / I_{k+},
6. relatively ambiguous ideals of Nk is #(I_{N}^{G} / I_{k}) = 5^{d},
7. (weakly) ambiguous classes of Nk is
#(Cl(N)^{G}) = 5^{d  1} h_{k} / (U_{k}:U_{k}∩Norm_{Nk}N^{*}) = 5^{d + q* 3},
where h_{k} = 1, 5^{q*} = (U_{k}∩Norm_{Nk}N^{*}:U_{k}^{5}), (U_{k}:U_{k}^{5}) = 5^{2} and
(U_{k}:U_{k}∩Norm_{Nk}N^{*}) = (U_{k}:U_{k}^{5}) / (U_{k}∩Norm_{Nk}N^{*}:U_{k}^{5}),
8. strongly ambiguous classes of Nk is #(I_{N}^{G} / P_{N}^{G}) = 5^{d  1} h_{k} / (U_{k}:Norm_{Nk}U_{N}),
where
I_{N}^{G} / P_{N}^{G} = I_{N}^{G} / I_{N}^{G}∩P_{N} ≅ I_{N}^{G}P_{N} / P_{N}.
Our first theorem extends
[Thm. IV (10), p. 481, Pa]
and
[Thm. 8, p. 223, PaWa1],
resp.
[PaWa2],
by giving the conductor f, the unit norm index
u = (U_{k}:Norm_{Nk}U_{N}),
and the differential principal factorization (DPF) type.
It is an analogue of [Thm., p. 8, Ho] for pure cubic fields.
Theorem 10.1.
The 5class group of the Galois closure N of a pure quintic field L = Q(D^{1/5}) is trivial,
Cl_{5}(N) = 1,
if and only if one of the following five conditions is satisfied.

Radicand D = 5, conductor f^{4} = 5^{6} (kind Ia), u = 1, DPF type θ

Radicand D = r with a prime r ≡ ±7 (mod 25), conductor f^{4} = r^{4} (kind II), u = 1, DPF type θ

Radicand D = q with a prime q ≡ ±2 (mod 5) but not ≡ ±7 (mod 25),
conductor f^{4} = 5^{2} q^{4} (kind Ib), u = 5, DPF type ε

Radicand D = q^{e1} 5^{e2} with a prime q ≡ ±2 (mod 5) but not ≡ ±7 (mod 25),
conductor f^{4} = 5^{6} q^{4} (kind Ia), u = 5, DPF type ε

Radicand D = q^{e1} r^{e2} ≡ ±1,±7 (mod 25)
with primes q,r ≡ ±2 (mod 5)
not both ≡ ±7 (mod 25), conductor f^{4} = q^{4} r^{4} (kind II), u = 5, DPF type ε
The exponent of the power in the unit index (U_{N}:U_{0}) = 5^{e}
takes the value e = 5 in all cases.
Furthermore, the 5class group of the pure quintic field L = Q(D^{1/5}) is also trivial, Cl_{5}(L) = 1.
Remark 10.1.
Theorem 10.1 deals with the cases where either τ = d = 1, q^{*} = 2 or τ = d = 2, q^{*} = 1
and consequently there are no ambiguous classes of Nk, since d + q^{*}  3 = 0.
Further, it is well known that 5  h_{N} if and only if 5  #(Cl(N)^{G}) and that 5  h_{L} implies 5  h_{N}.
The DPF types are enforced by s = σ = 0 and (P_{L}^{G} : P_{Q}) ≤ (I_{L}^{G} : I_{Q}) = 5^{τ}.
Examples 10.1.
Using our numerical results in section 8,
we give all radicands D ≤ 150 of pure quintic fields L = Q(D^{1/5})
where the various configurations of Theorem 10.1 actually occur.

A single prime divisor of the radicand D and of the conductor f:
D = r ≡ ±7 (mod 25) of second kind, f^{4} = r^{4}, for
D ∈ {7,43,107}.

A single prime divisor of the radicand D and two prime divisors of the conductor f:
D = q ≡ ±2 (mod 5) of first kind (Ib), f^{4} = 5^{2} q^{4}, for
D ∈ {2,3,13,17,23,37,47,53,67,73,83,97,103,113,127,137}.

Two prime divisors of the radicand D and of the conductor f:
D = q^{e1} 5^{e2} of first kind (Ia), f^{4} = 5^{6} q^{4}, for
D ∈ {10,15,20,40,45,65,75,80,85,115},
D = q^{e1} r^{e2} of second kind, f^{4} = q^{4} r^{4}, for
D ∈ {18,26,51,68,74}.
The next theorem extends those parts of
[Thm. IV (11), p. 481, Pa]
and
[Lem. 3.3, p. 204, Ii],
in conjunction with
[Thm. 4.3, p. 209, Ii],
where none of the prime divisors of the conductor f splits in k.
Theorem 10.2.
The 5class group of the Galois closure N of a pure quintic field L = Q(D^{1/5}) is cyclic of order 5,
Cl_{5}(N) = (5),
if and only if one of the following five conditions is satisfied
(for each condition, an integer m is defined).

Radicand D = r^{e1} 5^{e2} with a prime r ≡ ±7 (mod 25),
conductor f^{4} = 5^{6} r^{4} (kind Ia), m = 5, u = 5, DPF type η

Radicand D = q^{e1} r^{e2} ≡ ±1,±7 (mod 25)
with primes q,r ≡ ±7 (mod 25),
conductor f^{4} = q^{4} r^{4} (kind II), m = q, u = 5, DPF type η

Radicand D = q^{e1} r^{e2} not ≡ ±1,±7 (mod 25)
with primes q,r ≡ ±2 (mod 5)
and not q ≡ ±7 (mod 25), conductor f^{4} = 5^{2} q^{4} r^{4} (kind Ib), m = q, u = 25, DPF type γ

Radicand D = q^{e1} r^{e2} 5^{e3}
such that not q^{e1} r^{e2} ≡ ±1,±7 (mod 25), with primes q,r ≡ ±2 (mod 5)
and not q ≡ ±7 (mod 25), conductor f^{4} = 5^{6} q^{4} r^{4} (kind Ia), m = 5 q^{b}, u = 25, DPF type γ

Radicand D = q^{e1} r^{e2} s^{e3} ≡ ±1,±7 (mod 25)
with primes q,r,s ≡ ±2 (mod 5)
and not q,s ≡ ±7 (mod 25), conductor f^{4} = q^{4} r^{4} s^{4} (kind II), m = s q^{b}, u = 25, DPF type γ
and for none of these five conditions
(*) ε^{(r4  1) / 5} ≡ 1 (mod r),
where ε is a unit of F = Q(m^{1/5}) not contained in
< 1, E_{1}^{(1  ρ)3}, E_{2}^{5(1  ρ)2} >
and where
{ 1, E_{1}^{(1  ρ)3}, E_{2}^{(1  ρ)2} }
generates the unit group of F
and { E_{1}, E_{1}^{(1  ρ)}, E_{1}^{(1  ρ)2}, E_{2}, E_{2}^{(1  ρ)} }
generates the unit group
of the Galois closure C = Q(m^{1/5},ζ) of F modulo subfield units,
and Gal(Ck) = < ρ >.
The exponent of the power in the unit index (U_{N}:U_{0}) = 5^{e}
takes the value e = 6 in all cases.
Furthermore, the 5class group of the pure quintic field L = Q(D^{1/5}) is trivial, Cl_{5}(L) = 1.
Remark 10.2.
Theorem 10.2 deals with the cases where either τ = d = 2, q^{*} = 2 or τ = d = 3, q^{*} = 1
and consequently there are exactly 5 ambiguous classes of Nk, since d + q^{*}  3 = 1.
Examples 10.2.
Our numerical results in section 8 enable us
to give all radicands D ≤ 150 of pure quintic fields L = Q(D^{1/5})
where the various scenarios of Theorem 10.2 actually occur.

Two prime divisors of the radicand D and of the conductor f:
D = r^{e1} 5^{e2} of first kind (Ia), f^{4} = 5^{6} r^{4}, for
D ∈ {35}.

D = q^{e1} r^{e2} of second kind, f^{4} = q^{4} r^{4},
the first occurrence seems to be D = 301 = 7*43
and lies beyond the range covered by our computations.

Two prime divisors of the radicand D and three prime divisors of the conductor f:
D = q^{e1} r^{e2} of first kind (Ib), f^{4} = 5^{2} q^{4} r^{4}, for
D ∈ {6,12,14,21,28,34,39,46,48,52,56,63,69,86,91,92,94,
104,106,111,112,117,119,129,134,136,146,147,148}.
However, D = 141 is the smallest case where
Iimura's congruence condition (*) is satisfied by the unit ε
and we have e = 5, Cl_{5}(L) = (5), Cl_{5}(N) = (5,5,5,5), u = 5, DPF type ε.

Three prime divisors of the radicand D and of the conductor f:
D = q^{e1} r^{e2} 5^{e3} of first kind (Ia), f^{4} = 5^{6} q^{4} r^{4}, for
D ∈ {30,60,70,105,150},
D = q^{e1} r^{e2} s^{e3} of second kind, f^{4} = q^{4} r^{4} s^{4}, for
D ∈ {126}.
However, D = 140 is the smallest case where
Iimura's congruence condition (*) is satisfied by the unit ε,
and we have e = 5, Cl_{5}(L) = (5), Cl_{5}(N) = (5,5,5,5), u = 5, DPF type ε.
Note that D/5 ≡ ±1,±7 (mod 25) for D ∈ {90,120,130}.
Now we present a theorem containing a lot of new insights.
Theorem 10.3.
The 5class rank of the pure quintic field L = Q(D^{1/5}) is r_{5} Cl(L) ≥ 1 and
the 5class rank of the Galois closure N of L is r_{5} Cl(N) ≥ 2,
if one of the following conditions is satisfied.

Radicand D = r with a prime r ≡ +1 (mod 25), conductor f^{4} = r^{4} (kind II), u = 5, DPF type ζ_{1}

Radicand D = r with a prime r ≡ 1 (mod 25), conductor f^{4} = r^{4} (kind II), u = 5, DPF type ζ_{2}

Radicand D = l with a prime l ≡ +1 (mod 5) but not ≡ +1 (mod 25),
conductor f^{4} = 5^{2} l^{4} (kind Ib), u = 25, DPF type α_{2} or α_{1}

Radicand D = l with a prime l ≡ 1 (mod 5) but not ≡ 1 (mod 25),
conductor f^{4} = 5^{2} l^{4} (kind Ib), u = 5, DPF type δ_{2}, or u = 25, DPF type β_{2}
Examples 10.3.
Using our numerical results in section 8,
we give all radicands D ≤ 150 of pure quintic fields L = Q(D^{1/5})
where the various situations of Theorem 10.3 actually occur.

A single prime divisor of the radicand D and of the conductor f:
D = r ≡ +1 (mod 25) of second kind, f^{4} = r^{4}, for
D ∈ {101},
D = r ≡ 1 (mod 25) of second kind, f^{4} = r^{4}, for
D ∈ {149}.

A single prime divisor of the radicand D and two prime divisors of the conductor f:
D = l ≡ +1 (mod 5) of first kind (Ib), f^{4} = 5^{2} l^{4}, for
D ∈ {11,41,61,71,131} of type α_{2}, resp. D= 31 of type α_{1},
D = l ≡ 1 (mod 5) of first kind (Ib), f^{4} = 5^{2} l^{4}, for
D ∈ {19,29,59,79,89,109} of type δ_{2}, resp. D = 139 of type β_{2}.
Finally a result which recalls certain instances of
[Thm. 2.1, p. 412, Mo],
[Thm. 2.2, p. 413, Mo],
and
[Thm. V, p. 484, Pa].
Theorem 10.4.
1. If Cl_{5}(N) is cyclic, then 5 does not divide h_{L}.
2. If 5^{2} divides h_{N}, then 5 divides h_{L}.
3. If 5^{6} divides h_{N}, then 5^{2} divides h_{L}.
4. If 5 divides h_{L} exactly, then e ≥ 3, 5^{2} divides h_{N}, and Cl_{5}(N) is not cyclic.
5. If 5^{2} divides h_{L} and e ≥ 3, then 5^{6} divides h_{N}.
With regard to Thm. 10.4.5, we issue the following
Warning.
The pure quintic fields L = Q(D^{1/5}) with radicands D ∈ {31,33,88,131} show that
5^{2}  h_{L} and e ≤ 2 does not imply 5^{6}  h_{N}.
For D = 123, however, we actually see that the conditions
5^{2}  h_{L} and e = 3 imply 5^{6}  h_{N}, indeed.
A similar investigation of pure cubic fields.



Trade, Science, Art and Industry
Principal investigator of the
International Research Project
Towers of pClass Fields
over Algebraic Number Fields
supported by the Austrian Science Fund (FWF):
P 26008N25
