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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
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],
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].
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.
Trade, Science, Art and Industry
Principal investigator of the
International Research Project
Towers of p-Class Fields
over Algebraic Number Fields
supported by the Austrian Science Fund (FWF):