Fame For Styria 2014: Pure Quintic Fields

Pure Quintic Fields

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Pure Quintic Fields

1. Galois closure and subfield lattice.

Let q1,…,qs be pairwise distinct primes
such that s ≥ 1 and 5 may be among them.
Denote by L = Q(D1/5) the pure quintic number field
with fifth power free radicand D = q1e1 … qses,
where the exponents are integers 1 ≤ ej ≤ 4.
The field L is generated by adjoining the unique real solution
of the pure quintic equation X5 - D = 0 to the rational number field Q.
It is a non-Galois algebraic number field with signature (1,2)
and thus possesses four isomorphic complex fields
Lj = Q(ζjD1/5), 1 ≤ j ≤ 4,
all of whose arithmetical invariants coincide.

The normal closure of L is the compositum N = Q(D1/5,ζ)
of L = Q(D1/5) with the cyclotomic field k = Q(ζ),
where ζ = ζ5 = exp(2 π i / 5) denotes a primitive fifth root of unity.
N is a complex metacyclic field of degree 20 whose Galois group
Gal(N|Q) is the semidirect product M5 = C5 * C4 of two cyclic groups.

The cyclotomic field k is a complex cyclic quartic field
and contains the real quadratic field Q(51/2)
as its maximal real subfield k+ = Q(ζ + ζ-1).
Consequently, there exists a real intermediate field
M = Q(D1/5,51/2) of degree 10 between L and N,
which is non-Galois with signature (2,4).

2. Class numbers and unit index.

In 1973, Parry has determined the class number relation
55 hN = (UN:U0) hL4
[Thm. I, p. 476, Pa], where
UN denotes the unit group of N,
U0 is the subgroup of UN generated by
all units of the conjugate fields
Lj = Q(ζjD1/5), 0 ≤ j ≤ 4, of L and of k,
and the unit index (UN:U0) can take
seven possible values 5e with 0 ≤ e ≤ 6
[Thm. II, p. 478, Pa].

Examples 2.1.
Anticipating some of our numerical results in section 8,
we give the smallest radicands D of pure quintic fields L = Q(D1/5)
where the various values of the exponent e actually occur:
e = 6 for D = 6 = 2*3 of type γ,
e = 5 for D = 2 of type ε,
e = 4 for D = 22 = 2*11 of type β2,
e = 3 for D = 11 of type α2,
e = 2 for D = 31 of type α1,
e = 1 for D = 33 = 3*11 of type α2.
However, we point out that we did not find a realization of
e = 0,
that is, UN is never generated completely by subfield units.

At the end of his article, Parry suggests verbatim
"In conclusion the author would like to say that he believes
a numerical study of pure quintic fields would be most interesting"
[p. 484, Pa].
Of course, it would have been rather difficult
to realize Parry's desire in 1973.
But now, 40 years later, we are in the position to use
the powerful computer algebra systems PARI/GP and MAGMA
to start an attack against this hard problem.
This will be done in our present research line.

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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:

[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)

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