Fame For Styria 2014: Pure Quintic Fields 4

Pure Quintic Fields 4



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Pure Quintic Fields 4

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9. Algorithmic techniques.

  1. 1st step, the unit index (UN:U0):

    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 5-class groups Cl5(L), Cl5(M) and Cl5(N).
    Then we use Parry's class number relation to calculate the unit index
    (UN:U0) = 55 hN / hL4
    where hL = #Cl5(L) and hN = #Cl5(N).
    (This step executes rather quickly in about 3 hours of CPU time
    for the 90 fields with D ≤ 110 of section 8.)

  2. 2nd step, the ambiguous principal ideals:

    By means of MAGMA we compute all ideals which divide the relative different of N|k
    and perform a principal ideal test for each of them.
    (This step executes very rapidly, even for big radicands D > 100.)

  3. 3rd step, the unit norm index (Uk:NormN|kUN):

    Since the normal closure N of L is of signature (0,10),
    its torsion free unit rank is 0+10-1 = 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+4-1 = 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(N|k), then the number of all
4. absolutely ambiguous ideals of L|Q is #(ILG / IQ) = 5τ,
where ILGIk / Ik ≅ ILG / IL∩Ik = ILG / IQ,
5. intermediate ambiguous ideals of M|k+ is #(IMG / Ik+) = 5δ,
where IMGIk / Ik ≅ IMG / IM∩Ik = ILG / Ik+,
6. relatively ambiguous ideals of N|k is #(ING / Ik) = 5d,
7. (weakly) ambiguous classes of N|k is #(Cl(N)G) = 5d - 1 hk / (Uk:Uk∩NormN|kN*) = 5d + q*- 3,
where hk = 1, 5q* = (Uk∩NormN|kN*:Uk5), (Uk:Uk5) = 52 and
(Uk:Uk∩NormN|kN*) = (Uk:Uk5) / (Uk∩NormN|kN*:Uk5),
8. strongly ambiguous classes of N|k is #(ING / PNG) = 5d - 1 hk / (Uk:NormN|kUN),
where ING / PNG = ING / ING∩PN ≅ INGPN / PN.

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 = (Uk:NormN|kUN),
and the differential principal factorization (DPF) type.
It is an analogue of [Thm., p. 8, Ho] for pure cubic fields.

Theorem 10.1.
The 5-class group of the Galois closure N of a pure quintic field L = Q(D1/5) is trivial,
Cl5(N) = 1, if and only if one of the following five conditions is satisfied.
  1. Radicand D = 5, conductor f4 = 56 (kind Ia), u = 1, DPF type θ
  2. Radicand D = r with a prime r ≡ ±7 (mod 25), conductor f4 = r4 (kind II), u = 1, DPF type θ
  3. Radicand D = q with a prime q ≡ ±2 (mod 5) but not ≡ ±7 (mod 25),
    conductor f4 = 52 q4 (kind Ib), u = 5, DPF type ε
  4. Radicand D = qe1 5e2 with a prime q ≡ ±2 (mod 5) but not ≡ ±7 (mod 25),
    conductor f4 = 56 q4 (kind Ia), u = 5, DPF type ε
  5. Radicand D = qe1 re2 ≡ ±1,±7 (mod 25) with primes q,r ≡ ±2 (mod 5)
    not both ≡ ±7 (mod 25), conductor f4 = q4 r4 (kind II), u = 5, DPF type ε

The exponent of the power in the unit index (UN:U0) = 5e takes the value e = 5 in all cases.
Furthermore, the 5-class group of the pure quintic field L = Q(D1/5) is also trivial, Cl5(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 N|k, since d + q* - 3 = 0.
Further, it is well known that 5 | hN if and only if 5 | #(Cl(N)G) and that 5 | hL implies 5 | hN.
The DPF types are enforced by s = σ = 0 and (PLG : PQ) ≤ (ILG : IQ) = 5τ.

Examples 10.1.
Using our numerical results in section 8,
we give all radicands D ≤ 150 of pure quintic fields L = Q(D1/5)
where the various configurations of Theorem 10.1 actually occur.
  1. A single prime divisor of the radicand D and of the conductor f:
    D = r ≡ ±7 (mod 25) of second kind, f4 = r4, for
    D ∈ {7,43,107}.
  2. 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), f4 = 52 q4, for
    D ∈ {2,3,13,17,23,37,47,53,67,73,83,97,103,113,127,137}.
  3. Two prime divisors of the radicand D and of the conductor f:
    D = qe1 5e2 of first kind (Ia), f4 = 56 q4, for
    D ∈ {10,15,20,40,45,65,75,80,85,115},
    D = qe1 re2 of second kind, f4 = q4 r4, 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 5-class group of the Galois closure N of a pure quintic field L = Q(D1/5) is cyclic of order 5,
Cl5(N) = (5), if and only if one of the following five conditions is satisfied
(for each condition, an integer m is defined).
  1. Radicand D = re1 5e2 with a prime r ≡ ±7 (mod 25),
    conductor f4 = 56 r4 (kind Ia), m = 5, u = 5, DPF type η
  2. Radicand D = qe1 re2 ≡ ±1,±7 (mod 25) with primes q,r ≡ ±7 (mod 25),
    conductor f4 = q4 r4 (kind II), m = q, u = 5, DPF type η
  3. Radicand D = qe1 re2 not ≡ ±1,±7 (mod 25) with primes q,r ≡ ±2 (mod 5)
    and not q ≡ ±7 (mod 25), conductor f4 = 52 q4 r4 (kind Ib), m = q, u = 25, DPF type γ
  4. Radicand D = qe1 re2 5e3 such that not qe1 re2 ≡ ±1,±7 (mod 25), with primes q,r ≡ ±2 (mod 5)
    and not q ≡ ±7 (mod 25), conductor f4 = 56 q4 r4 (kind Ia), m = 5 qb, u = 25, DPF type γ
  5. Radicand D = qe1 re2 se3 ≡ ±1,±7 (mod 25) with primes q,r,s ≡ ±2 (mod 5)
    and not q,s ≡ ±7 (mod 25), conductor f4 = q4 r4 s4 (kind II), m = s qb, 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(m1/5) not contained in < -1, E1(1 - ρ)3, E25(1 - ρ)2 >
and where { -1, E1(1 - ρ)3, E2(1 - ρ)2 } generates the unit group of F
and { E1, E1(1 - ρ), E1(1 - ρ)2, E2, E2(1 - ρ) } generates the unit group
of the Galois closure C = Q(m1/5,ζ) of F modulo subfield units,
and Gal(C|k) = < ρ >.

The exponent of the power in the unit index (UN:U0) = 5e takes the value e = 6 in all cases.
Furthermore, the 5-class group of the pure quintic field L = Q(D1/5) is trivial, Cl5(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 N|k, 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(D1/5)
where the various scenarios of Theorem 10.2 actually occur.
  1. Two prime divisors of the radicand D and of the conductor f:
    D = re1 5e2 of first kind (Ia), f4 = 56 r4, for
    D ∈ {35}.
  2. D = qe1 re2 of second kind, f4 = q4 r4,
    the first occurrence seems to be D = 301 = 7*43
    and lies beyond the range covered by our computations.
  3. Two prime divisors of the radicand D and three prime divisors of the conductor f:
    D = qe1 re2 of first kind (Ib), f4 = 52 q4 r4, 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, Cl5(L) = (5), Cl5(N) = (5,5,5,5), u = 5, DPF type ε.
  4. Three prime divisors of the radicand D and of the conductor f:
    D = qe1 re2 5e3 of first kind (Ia), f4 = 56 q4 r4, for
    D ∈ {30,60,70,105,150},
    D = qe1 re2 se3 of second kind, f4 = q4 r4 s4, 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, Cl5(L) = (5), Cl5(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 5-class rank of the pure quintic field L = Q(D1/5) is r5 Cl(L) ≥ 1 and
the 5-class rank of the Galois closure N of L is r5 Cl(N) ≥ 2,
if one of the following conditions is satisfied.
  1. Radicand D = r with a prime r ≡ +1 (mod 25), conductor f4 = r4 (kind II), u = 5, DPF type ζ1
  2. Radicand D = r with a prime r ≡ -1 (mod 25), conductor f4 = r4 (kind II), u = 5, DPF type ζ2
  3. Radicand D = l with a prime l ≡ +1 (mod 5) but not ≡ +1 (mod 25),
    conductor f4 = 52 l4 (kind Ib), u = 25, DPF type α2 or α1
  4. Radicand D = l with a prime l ≡ -1 (mod 5) but not ≡ -1 (mod 25),
    conductor f4 = 52 l4 (kind Ib), u = 5, DPF type δ2, or u = 25, DPF type β2


Examples 10.3.
Using our numerical results in section 8,
we give all radicands D ≤ 150 of pure quintic fields L = Q(D1/5)
where the various situations of Theorem 10.3 actually occur.
  1. A single prime divisor of the radicand D and of the conductor f:
    D = r ≡ +1 (mod 25) of second kind, f4 = r4, for
    D ∈ {101},
    D = r ≡ -1 (mod 25) of second kind, f4 = r4, for
    D ∈ {149}.
  2. 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), f4 = 52 l4, for
    D ∈ {11,41,61,71,131} of type α2, resp. D= 31 of type α1,
    D = l ≡ -1 (mod 5) of first kind (Ib), f4 = 52 l4, for
    D ∈ {19,29,59,79,89,109} of type δ2, resp. D = 139 of type β2.

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 Cl5(N) is cyclic, then 5 does not divide hL.
2. If 52 divides hN, then 5 divides hL.
3. If 56 divides hN, then 52 divides hL.
4. If 5 divides hL exactly, then e ≥ 3, 52 divides hN, and Cl5(N) is not cyclic.
5. If 52 divides hL and e ≥ 3, then 56 divides hN.

With regard to Thm. 10.4.5, we issue the following

Warning.
The pure quintic fields L = Q(D1/5) with radicands D ∈ {31,33,88,131} show that
52 | hL and e ≤ 2 does not imply 56 | hN.
For D = 123, however, we actually see that the conditions
52 | hL and e = 3 imply 56 | hN, indeed.

A similar investigation of pure cubic fields.



Town Hall Graz, Figures
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:

[Ho] T. Honda,
Pure cubic fields whose class numbers are multiples of three,
J. Number Theory 3 (1971), no. 1, 7 - 12.

[Ii] K. Iimura,
A criterion for the class number of a pure quintic field to be divisible by 5,
J. reine angew. Math. 292 (1977), 201 - 210.

[MAGMA] The MAGMA Group,
MAGMA Computational Algebra System, Version 2.21-1,
Sydney, 2014, http://magma.maths.usyd.edu.au

[Mo] R. A. Mollin,
Class numbers of pure fields,
Proc. Amer. Math. Soc. 98 (1986), no. 3, 411 - 414.

[PARI] The PARI Group,
PARI/GP, Version 2.7.2
Bordeaux, 2014, http://pari.math.u-bordeaux.fr

[Pa] C. J. Parry,
Class number relations in pure quintic fields,
Symposia Mathematica 15 (1975), 475 - 485, Academic Press, London.
(Convegno di Strutture in Corpi Algebrici, INDAM, Rome, 1973)

[PaWa1] C. J. Parry and C. D. Walter,
The class number of pure fields of prime degree,
Mathematika 23 (1976), 220 - 226.

[PaWa2] C. J. Parry and C. D. Walter,
Corrigendum: The class number of pure fields of prime degree,
Mathematika 24 (1977), 133.

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